the mean value for infinite volume measures, infinite...

Research ArticleThe Mean Value for Infinite Volume MeasuresInfinite Products and Heuristic Infinite DimensionalLebesgue Measures

Jean-Pierre Magnot

Lycee Jeanne drsquoArc Avenue de Grande Bretagne 63000 Clermont-Ferrand France

Correspondence should be addressed to Jean-Pierre Magnot jpmagnotgmailcom

Received 1 July 2016 Revised 16 September 2016 Accepted 19 September 2016 Published 30 January 2017

Academic Editor Tepper L Gill

Copyright copy 2017 Jean-Pierre MagnotThis is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

One of the goals of this article is to describe a setting adapted to the description of means (normalized integrals or invariant means)on an infinite product of measured spaces with infinite measure and of the concentration property on metric measured spacesinspired from classical examples of means In some cases we get a linear extension of the limit at infinity Then the mean value onan infinite product is defined first for cylindrical functions and secondly taking the uniform limit Finally the mean value for theheuristic Lebesgue measure on a separable infinite dimensional topological vector space (eg on a Hilbert space) is defined Thislast object which is not the classical infinite dimensional Lebesgue measure but its ldquonormalizedrdquo version is shown to be invariantunder translation scaling and restriction

1 Introduction

The very early starting point of this work is the well-known lack of adequate definition of an infinite dimensionalLebesgue measure on a Hilbert space Even if infinite dimen-sional version of the Lebesgue measure is well known onHilbert space [1 2] and translation invariant measures arealready described in Banach spaces these measures fail tohave ldquoenoughrdquo measurable sets with finite nonzero measureFrom another approach some expressions of the type

lim 1120583 (119880119899) int119880119899 119891119889120583 (1)

are well known since [3] see for example [4 5] onR119899 andwefind similar expressions in the theory of infinite dimensionaloscillatory integrals see for example [6ndash10] Yet in anothersetting there is a property of concentration of measure inmetric measured spaces which can coincide with the defini-tion of a mean for uniform functions in for example Sinfinsee for example [11ndash13] These approaches except the onesinvolving infinite dimensionalmeasures appear as relevant ofthe same procedure defining means from limits of measuresThis is why following [14] we suggest a setting in Section 2 for

means defined by limits of finite measures To our knowledge(and surprisingly also) these frameworks have not beengathered yet We show how what we defined as Dirac meansin [14] or their generalizations that we call probability meansor limit means describe a unified framework to deal withconcentration properties in metric measured spaces on onehand and integrals of cylindrical functions on the other hand

The theory developed in Section 2 is then specialized toa restricted class of means first to the means obtained witha 120590-finite Radon measure using a creasing sequence (119880119899)119899isinNof Borel subsets with finite measure satisfying ⋃119899isinN119880119899 = 119883among other technical conditions by

119891 = lim119899rarr+infin

1120583 (119880119899) int119880119899 119891119889120583 (2)

following definitions present in [3ndash5] We give in a way assystematic as possible their basic properties in Section 3Since this mean value depends (in general) on the sequence119880 and on the measure 120583 we do not adopt the notation 119891but prefer WMV119880

120583 (119891) or MV120583(119891) abbreviations for ldquoweakmean valuerdquo and for ldquomean valuerdquo Formulas for changing ofmeasure lead us to an extension of the asymptotic comparison

HindawiJournal of MathematicsVolume 2017 Article ID 9853672 14 pageshttpsdoiorg10115520179853672

2 Journal of Mathematics

of functions (119891siminfin 119892 119891 = 119874(119892) and 119891 = 119900(119892)) tomeasures As a particular case the mean value with respectto the Lebesgue measure on R appears as a linear extensionof the limit at infin of functions We know very few aboutthe behaviour of the mean value of limit of functions themean value is not continuous for vague convergence butcontinuous for uniform convergence There is certainly anintermediate kind of convergence more adapted to meanvalues to be determined We also give an application of thisnotion the homology map as a mean value of a function onthe space of harmonic forms using Hodge theory

Secondly we get to infinite products of measured spacesin Section 4 Recall that there is an induced measure on aninfinite product of measured spaces only if we have spaceswith finite measures We consider cylindrical functions anddefine very easily their mean values as mean values offunctions defined on a finite product of measured spaceThen we extend it to functions that are uniform limits ofsequences of cylindrical functions As an application we givea definition of themean value on infinite configuration spacesfor Poisson measure

Finally we get to vector subspaces of Hilbert spacesin Section 4 This is where we decide to focus on theannounced heuristic infinite dimensional Lebesgue meanwhich is not the infinite dimensional Lebesgue measuredescribed in [1 2]Themean value is developed and we studyits invariance properties It appears invariant by translationand by scaling and also by action of the unitary groupBut the last one remains dependent on the choice of theorthonormal basis used for the definition which is classicalin the procedure of approximation by cylindrical functions[15] As a concluding remark we show that this approachhas a technical difference with the approach by measureson infinite dimensional spaces We show that the meanvalue of a bounded continuous function 119891 remains the samewhile restricting to a dense vector subspace This exhibits astriking difference from for example the Wiener measureon continuous paths for which the space of 1198671 paths is ofmeasure 0With all these elements we can now explainwhereis the originality of our approachHere the total volume is notconsidered as a constant of the total space but as a scale-likeelement to compare with the integral of a function

2 The Space of Means Spanned by Sequencesof Finite Measures

Let (119883 120583) be a measured space Following [5 13] let us fix avector subspaceF sub 119871infin(119883 120583) such that 1119883 isin FAmean onF is a linearmap120601 Frarr C such that120601(1119883) = 1Alternatelyif (119883 119889) is a metric space given F sub 1198620

119887(119883) (space ofcontinuous bounded maps) a mean on F is a linear map120601 F rarr C such that 120601(1119883) = 1 These two terminologiescome from the basic example where 120583 is a Borel probabilitymeasure on a metric space (119883 119889) for which the mean of acontinuous integrable map 119891 is its expectation value

int119883119891119889120583 (3)

and can be approximated by sequences of barycenters ofDirac measures via Monte Carlo methods

LetF0 be a vector space of (bounded measurable) mapsthat contains constant maps with values in a complete locallyconvex topological vector space (clctvs) 119881 We shall call allalong this paper M(119883F0) the space of means that is thespace of linear maps 120601 F0 rarr R such that 120601(1119883) = 1Themean 120601 can be defined on another domain 119863 sup F0 but thespace F0 will serve as reference domain Moreover we setK = R or C

21 Means Spanned by Probability Measures Let 119883 be acomplete metric space and let1198620

119887(119883) be the space of boundedK-valued continuous maps on119883We note by P(119883) the spaceof Borel probability measures on119883 Let us first set 119881 = K

Definition 1 A K-probability mean is a linear map 120591 D120591 sub1198620119887(119883) rarr K which is defined as the limit of barycenters with

K-weights of a sequence of Borel probability measures on119883 that isexist (120583119899 120572119899)119899isinN isin (P (119883) times K)N forall119898 isin N

lowast 119898sum119899=0

120572119899 = 0 and forall119891 isin 1198620119887 (119883) 120591 (119891)

= lim119898rarr+infin

1sum119898119899=0 120572119899 (

119898sum119899=0

120572119899120583119899 (119891)) (4)

We note by PMK(119883) the space of K-probability meansby PMK(119883) the set of probability means 120591 such that D120591 =1198620119887(119883) and by PM

+

R(119883) the means 120591 obtained by a sequence(120572119899)119899isinN isin Rlowast+ and we set

PM+R (119883) = PMC (119883) cap PM

+

R (119883) (5)

We have a special class spanned by the Dirac measures

Definition 2 (see [14]) A K-Dirac mean is a linear map 120591 D120591 sub 1198620

119887(119883) rarr Kwhich is defined as the limit of barycenterswith K-weights of a sequence of Dirac measures on119883 119898sum119899=0

120572119899 = 0 and forall119891 isin 1198620119887 (119883) 120591 (119891)


1sum119898119899=0 120572119899 (

119898sum119899=0

120572119899120575119909119899 (119891)) exist (119909119899 120572119899)119899isinN isin (119883 times K)N forall119898 isin N

lowast(6)

Wenote by DMK(119883) DMK(119883) DM+

R(119883) DM+R(119883)

the sets of Dirac means corresponding respectively toPMK(119883) PMK(119883) PM

+

R(119883) PM+R(119883)

Proposition 3 PMK(119883) PMK(119883) DMK(119883) andDMK(119883) are K-affine spaces

Journal of Mathematics 3

The proof is obvious adapting elementary proofs on(classical finite) barycenters We give also the following inorder to make quickly the link with theMonte Carlo method

Proposition 4 If 119883 is moreover a locally compact manifoldone has the following inclusions

(i) P(119883) sub DM+R(119883)

(ii) If119883 is compact P(119883) = DM+R(119883) = PM+

R(119883)Proof (i) Let 120583 isin P(119883) and let (119909119899)119899isinN be a uni-formly distributed sequence with respect to 120583 Then forall119891 isin1198620119887(119883) lim119898rarr+infin(1(119899 + 1))sum119898

119899=0 119891(119909119899) = E120583(119891) = int119883 119891119889120583Thus

120583 (119891) = lim119898rarr+infin

1119899 + 1119898sum119899=0

120575119909119899 (119891) (7)

(ii) If 119883 is compact the space of (signed) finite measureson 119883 coincides with (1198620

119887(119883))1015840 Since PMR(119883) sub (1198620119887(119883))1015840

we get the result

22 Probability Means in the mm-Space Setting We use twohandbooks for preliminaries on these notions [11 13]

Definition 5 (see [12]) A space with metric and measure ora metric measured space (mm-space for short) is a triple(119883 119889 120583) where (119883 119889) is a metric space and 120583 is a probabilitymeasure on the Borel tribu on119883

Let 119860 sub 119883 and let 120576 gt 0We note

119860120576 = 119909 isin 119883 | 119889 (119860 119909) lt 120576 (8)

Definition 6 (see [12]) A Levy family is a sequence X =(119883119899 119889119899 120583119899)119899isinN of mm-spaces if for each sequence (119860119899)119899isinN119860119899 is a Borel subset of 119883119899 forall119899 isin N (9)

satisfying

lim inf119899rarr+infin

120583119899 (119860119899) gt 0 (10)

and then

lim119899rarr+infin

120583119899 ((119860119899)120576) = 1 forall120576 gt 0 (11)

In the sequel we shall assume that

119883119899 sub 119883119899+1 forall119899 isin N (12)

with continuous injection Notice that we do not assume that119889119899 is the restriction of 119889119899+1 which allows us some freedomon metric requirements The technical necessary conditionis the following let 119899 isin N and let 119861119899+1 be a Borel subset of119883119899+1Then 119861119899+1 cap 119883119899 is a Borel subset of 119883119899We have herea priori a class of limit means following the terminology ofDefinition 11 Let us quote first the classical (and historical)example of a Levy family see for example [11] section 3 1

219

which gives an example of mean value

Example 7 (the Levy family of spheres and the concentrationphenomenon) Let us consider the sequence of inclusions

1198781 sub 1198782 sub sdot sdot sdot sub 119878119899 sub 119878119899+1 sdot sdot sdot sub 119878infin = infin⋃119899=1

119878119899 (13)

equipped with the classical Euclide (or Hilbert) distance and(except for 119878infin) the normalized spherical measure 120583 (we dropthe index for themeasure in sake of clear notations)Then forany R-valued 1-Lipschitz function on 119878infin there exists 119886 isin R

such that

120583 119909 isin 119878119899 | 1003817100381710038171003817119891 (119909) minus 1198861003817100381710038171003817 gt 120598 lt 2119890minus(119899minus1)12059822 forall120598 gt 0 (14)

In a more intuitive formation one can say that any 1-Lipschitz function concentrates around a real value 119886 withrespect to 120583 We leave the reader with [11] for more on themetric geometry of this example

Proposition 8 Let X = (119878119899 sdot 120583)119899isinNlowast Then for any 1-Lipschitz function119891defined on 119878infin andwith the notations usedbefore

119871119872X (119891) = 119886 (15)

Example 9 For Levy families induced by Lebesgue measureslet 119898 119899 isin (Nlowast)2 Take 119870119898 sub R119899 For each 119898 isin Nlowast we equip119870119898 with the usual distance 119889 induced by R119899 and with theprobability measure

120583119899 = 1119870119899120582 (119870119899)120582 (16)

SettingK = (119870119898 119889 120583119898)119898isinNlowast we get thatK is a Levy familybut there is no concentration property This example will bestudied in the next sections of this article

Definition 10 Let 119891 119883 rarr C be a map such that for each119899 isin N the restriction of 119891 to 119883119899 is 120583119899-integrable Then themean value of 119891 with respect to the familyX is

WMVX (119891) = lim119899rarr+infin

int119883119899

119891119889120583119899 (17)

if the limit exists

23 Limit Means and Infinite Dimensional Integrals

Definition 11 Let 119883 = (119883119899 120591119899)119899isinN be a sequence of probabil-ity spaces such that

(i) forall119899 isin N 119883119899 is a metric space(ii) forall119899 isin N 119883119899 sub 119883119899+1 and the topology of 119883119899+1

restricted to119883119899 coincides with the topology of119883119899

(iii) forall119899 isin N 120591119899 isin PMC(119883119899) Then we define for themaps 119891 defined on ⋃119899isinN119883119899 if forall119899 isin N 119891|119883119899 isin D120591119899and if the limit converges

LM119883 (119891) = lim119899rarr+infin

120591119899 (119891) (18)

called limit mean of 119891 with respect to119883


This definition intends to fit with the procedure ofintegration of cylindrical functions in Hilbert spaces Let usfirst describe the ldquotoyrdquo example of these infinite dimensionalintegrals where such an approach is not needed the limitmean considered is in fact a Dirac mean

Example 12 (Daniell integral) Let us consider cylinder func-tions 119891 on [0 1]N Let

119875119896 (1199091 1199092 ) isin RN 997891997888rarr (1199091 1199092 119909119896 0 ) (19)

be the 119896-dimensional projection Then there exists 119897 isin Nlowast

such that

119891 = 119891 ∘ 119875119897 (20)

Then adequate sequences for the Monte Carlo methodare those whose push-forwards on [0 1]119896 are also adequatefor this method The projectors 119875119896 converge (weakly) toidentity the condition on the sequence (119909119899) is that for each119896 isin N the push-forwards of the sequences (119875119896(119909119899)) on [0 1]119896fit with the desired conditions the sequence (119875119896(119909119899)) is aMonte Carlo sequence for the cube [0 1]119896 equipped with the(trace of) Lebesgue measure It is well known that such asequence (119909119899) exists through for example the powers of 120587119909119899 = (119899120587119897+1 minus int (119899120587119897+1))

119897isinNisin [0 1]N forall119899 isin N

lowast (21)

where int(119909) is the integer part of the real number 119909 ThusDaniell integral appears by its definition as a limit mean forthe sequence (119883119899)Nlowast defined by 119883119899 = [0 1]119899 equipped withthe classical Lebesque measure But Daniell integral appearsalso as a Dirac mean whose domain contains cylindricalfunctions

This example motivates the comparison between limitmeans PMK(119883) and DMK(119883)Example 13 (Fresnel-type integrals) First let 120591 isin PMK(119883)There exists a sequence (120583119899)N isin P(119883)N satisfyingDefinition 1and for each 119896 isin Nlowast there exists a sequence of Diracmeasures (120575(119896)119899 )119899isinN which converge to 120583119896 with respect to theMonte Carlo method Thus finding a sequence of Diracmeasures which can define a Dirac mean which coincideswith 120591 becomes a problem of extracting a sequence of Diracmeasures which converges to 120591The same is for a limitmean 120591= ldquolim 120591119899rdquo Let us describe more precisely the open problemson the example of oscillatory and Fresnel integrals Let Φ isin119862infin(R119899R) be a fixed function Following [16] (see eg [7 817 18]) we define the following

Definition 14 Let 119891 be a measurable function onR119899 Let 120593 isinS(R119899) be a weight function such that 120593(0) = 1 If the limit

lim120598rarr0

intR119899119890119894Φ(119909)119891 (119909) 120593 (120598119909) 119889119909 (22)

exists and is independent of the fixed function 120593 then thislimit is called oscillatory integral of119891with respect toΦ notedby

int119900

R119899119890119894Φ(119909)119891 (119909) 119889119909 (23)

The choice Φ(119909) = (1198942ℎ)|119909|2 is of particular interest andis known under the name of Fresnel integralThis choice givesus amean up to normalization by a factor (2119894120587ℎ)minus1198892 and canbe generalized to a Hilbert spaceH the following way

Definition 15 A Borel measurable function 119891 H rarr C iscalled ℎ-integrable in the sense of Fresnel for each creasingsequence of projectors (119875119899)119899isinN such that lim119899rarr+infin119875119899 = IdHand the finite dimensional approximations of the oscillatoryintegrals of 119891

int119900

Im119875119899

119890(1198942ℎ)|119875119899(119909)|2119891 (119875119899 (119909)) 119889 (119875119899 (119909))sdot int119900

Im119875119899

119890(1198942ℎ)|119875119899(119909)|2119889 (119875119899 (119909))minus1(24)

are well defined and the limit as 119899 rarr +infin does not dependon the sequence (119875119899)119899isinN In this case it is called infinitedimensional Fresnel integral of 119891 and noted by

int119900

H

119890(1198942ℎ)|119909|2119891 (119909) 119889 (119909) (25)

The invariance under the choices of the map 120593 andthe projections 119875119899 is assumed mostly to enable strongeranalysis on these objects which intend to be useful todescribe physical quantities and hence can be manipulated inapplications where one sometimes works ldquowith no fear on themathematical rigorrdquo in calculations But we can also remarkthat

(i) for functions 119891 defined on R119899 the map

119891 997891997888rarr int119900

R119899119890119894Φ(119909)119891 (119909) 119889119909 isin PMC (R119899) (26)

(ii) the map

119891 997891997888rarr lim119899rarr+infin

int119900

R119899119890119894Φ(119909)119891 (119909) 119889119909 (27)

is a limit mean through the sequence R sub sdot sdot sdot sub R119899 subR119899+1 sub sdot sdot sdot subH

The limit mean obtained is got through the classical trick ofcylindrical functions which we shall also use in the sequelBut we have no way to define some adequate sequenceof Dirac means which could approximate the oscillatoryintegral even in the finite dimensional case

Following another approach from [14] one can try to giveanother definition to the oscillatory integral straightway Letus begin with integration of cylinder functions on the infinitecube [0 1]N (Daniell integral) as described in the previous


example Let us consider now cylinder functions 119890119894Φ sdot 119891 on[0 1]N Let 119875 be a finite dimensional projection such that

119891 = 119891 ∘ 119875 (28)

and let

119878119875 = 119878 ∘ 119875 (29)

Then adequate sequences for the Monte Carlo methodare those whose push-forwards on [0 1]dim Im119875 are alsoadequate for this method Taking now a creasing sequenceof orthogonal projectors 119875119896 converging (weakly) to identitythe condition on the sequence (119909119899) is that for each 119896 isin N thepush-forwards of the sequences (119875119896(119909119899)) on [0 1]dim Im119875119896 fitwith the desired conditions the sequence (119875119896(119909119899)) is a MonteCarlo sequence for the cube [0 1]dim Im119875119896 equipped with the(trace of) Lebesgue measure It is well known that such asequence (119909119899) exists through for example the powers of 120587119909119899 = (119899120587119897+1 minus int (119899120587119897+1))


lowast (30)

where int(119909) is the integer part of the real number 119909 Let usnow fix 119889 = dim Im119875119896Themaps 120595 defined by for example

119909 = (1199091 119909119889) isin R119889 997891997888rarr 120595 (119909)

= 1radic2120587 (int1199091

minusinfin119890119905211198891199051 int119909119889

minusinfin1198901199052119889119889119905119899)

(31)

or

119909 = (1199091 119909119889) isin R119889 997891997888rarr 120595 (119909)

= (Argth (1199091) Argth (119909119889))(32)

are diffeomorphisms R119889 rarr ]0 1[119889 sub [0 1]119889 and we get

intR119899119890119894Φ(119909)119891 (119909) 120593 (120598119909) 119889119909 = int

R119899119890119894Φ∘120595minus1(119909)119891 ∘ 120595minus1 (119909) 120593

∘ 120595minus1 (120598119909) 10038161003816100381610038161003816det (119863119909120595minus1)10038161003816100381610038161003816 119889119909(33)

thus a pull-back of the Monte Carlo method can be per-formed this way to get a Dirac mean which could be consid-ered as an oscillatory integral But nothing can ensure to ourknowledge that this approach defines the same oscillatoryintegral as Definition 14

3 Example Mean Value on a Measured Space

On measured spaces the definitions are those given inclassical mathematical literature see for example [3ndash5]

31 Definitions On measured spaces the definitions arethose given in classical mathematical literature see for exam-ple [3] Let (119883 120583) be a topological space equipped with a 120590-additive positive measure 120583 Let T(119883) be a Borel 120590-algebra

on119883We note byRen120583(119883) the set of sequences119880 = (119880119899)119899isinN isinT(119883)N such that

(1) ⋃119899isinN119880119899 = 119883(2) forall119899 isin N 0 lt 120583(119880119899) lt +infin and 119880119899 sub 119880119899+1

Remark 16 We have in particular lim119899rarr+infin120583(119880119899) = 120583(119883)In what follows we assume the natural condition Ren120583 =0

Definition 17 Let 119880 isin Ren120583 Let 119881 be a separable completelocally convex topological vector space (sclctvs) Let 119891 119883 rarr119881 be ameasurablemapWe define if the limit exists the weakmean value of 119891 with respect to 119880 as

WMV119880120583 (119891) = lim

119899rarr+infin

1120583 (119880119899) int119880119899 119891119889120583 (34)

Moreover if WMV119880120583 (119891) does not depend on 119880 we call it

mean value of 119891 noted by MV120583(119891)Notice that

(i) if 119881 = R setting 119891+ = (12)(119891 + |119891|) and 119891minus = (12)(119891minus |119891|) WMV119880120583 (119891) =WMV119880

120583 (119891+) +WMV119880120583 (119891minus)

for each119880 isin Ren120583 if 119891 119891+ and 119891minus have a finite meanvalue

(ii) The sameway if119881 = C WMV119880120583 (119891) =WMV119880

120583 (R119891)+119894WMV119880120583 (I119891) for each 119880 isin Ren120583

(iii) We denote by F119880120583 the set of functions 119891 such that

WMV119880120583 (119891) exists in 119881 and byF120583 the set of functions119891 such that MV120583(119891) is well defined

Examples 1 (1) Let (119883 120583) be an arbitrary measured spaceLet 119891 = 1119883 Let 119880 isin Ren120583 forall119899 isin N (1120583(119880119899)) int119880119899 119891119889120583 =120583(119880119899)120583(119880119899) = 1 so that

MV120583 (1119883) = 1 (35)

(2) Let (119883 120575119909) be a space 119883 equipped with the Diracmeasure at 119909 isin 119883 Let 119891 be an arbitrary map to an arbitrarysclctvs 119880 isin Ren120575119909 hArr forall119899 isin N 120575119909(119880119899) gt 0 hArr forall119899 isin N 119909 isin119880119899Thus if 119880 isin Ren120575119909 forall119899 isin N (1120575119909(119880119899)) int119880119899 119891119889120575119909 = 119891(119909)so that

MV120575119909(119891) = 119891 (119909) (36)

(3) Let (119883 120583) be ameasured spacewith120583(119883) lt +infin Let119891be an arbitrary boundedmeasurablemapThen one can showvery easily that we recover the classical mean value of 119891

MV120583 (119891) = 1120583 (119883) int119883 119891119889120583 (37)

(4) Let119883 = R equippedwith the classical Lebesguemeas-ure 120582 Let 119892 isin 1198711(RR+) (integrable R+-valued function)


Let 119880 isin Ren120582 We have that lim119899rarr+infin int119880119899119892119889120582 le int

R119892119889120582 lt+infin so that

MV120582 (119892) = 0 (38)

(5) Let 119883 = R equipped with the Lebesgue measure120582 Let 119891(119909) = sin(119909) and let 119880119899 = [minus(119899 + 1) (119899 + 1)]The map sin is odd so that WMV119880

120582 (sin) = 0 Now let1198801015840119899 = [minus2120587119899 2120587119899] cup ⋃119899

119895=0[2(119899 + 119895)120587 (2(119899 + 119895) + 1)120587] ThenWMV1198801015840

120582 (sin) = 15120587 This shows that sin has no (strong)mean value for the Lebesgue measure

(6) Let119883 = N equippedwith 120574 the countingmeasure Let119899 isin N and set119880119899 = [0 119899]capN Let (119906119899) isin RN and119880 = (119880119899)119899isinNThen

WMV119880120574 (119906119899) = lim

119899rarr+infin

1119899 + 1119899sum

119896=0

119906119896 (39)

is the Cesaro limit

32 Basic Properties In what follows and till the end of thispaper we assume the natural condition Ren120583 = 0 for themeasures 120583 we considerProposition 18 Let (119883 120583) be ameasured space Let119880 isin 119877119890119899120583Then

(1) F119880120583 is a vector space and119882119872119881119880

120583 is linear

(2) F120583 is a vector space and119872119881120583 is linearProof The proof is obvious

We now clarify the preliminaries that are necessary tostudy the perturbations of the mean value of a fixed functionwith respect to perturbations of the measure

Proposition 19 Let 120583 and ] be Radon measures Let119880 = (119880119899)119899isinN isin 119877119890119899120583 cap 119877119890119899] Assume that Θ(120583 ]) =lim119899rarr+infin(120583(119880119899)(120583 + ])(119880119899)) isin [0 1] exists Then 119891 isin F119880

120583+]and

119882119872119881119880120583+] (119891) = Θ (120583 ])119882119872119881119880

120583 (119891)+ Θ (] 120583)119882119872119865119880] (119891) (40)

Proof

1(120583 + ]) (119880119899) int119880119899 119891119889 (120583 + ])= 120583 (119880119899)(120583 + ]) (119880119899)

1120583 (119880119899) int119880119899 119891119889120583+ ] (119880119899)(120583 + ]) (119880119899)

1] (119880119899) int119880119899 119891119889]

(41)

Thus we get the result taking the limit

Proposition 20 Let 120583 be a Radon measure and let 119896 isin Rlowast+

Then 119877119890119899119896120583 = 119877119890119899120583 F119896120583 = F120583 moreover forall119880 isin 119877119890119899120583F119880

119896120583 = F119880120583 and119882119872119881119880

119896120583 = 119882119872119865119880120583 Proof The proof is obvious

Theorem 21 Let 120583 be a measure on 119883 let 119880 isin 119877119890119899120583 and119891 isin F119880120583 Let

M (120583 119880 119891)= ] | 119880 isin 119877119890119899] 119882119872119881119880

] (119891) = 119882119872119881119880120583 (119891) (42)

M(120583 119880 119865) is a convex coneProof Let 119896 gt 0 and let ] isin M(120583 119880 119865) Setting ]1015840 = 119896]we get WMV119880

]1015840(119891) = WMV119880] (119891) by Proposition 20 thus

M(120583 119880 119865) is a coneNow let (] ]1015840) isin M(120583 119880 119865)2 Let 119905 isin [0 1] and let ]10158401015840 =119905] + (1 minus 119905)]1015840(i) Let us show that 119880 isin Ren]10158401015840 Let 119899 isin N We have

]10158401015840(119880119899) = 119905](119880119899) + (1 minus 119905)]1015840(119880119899) so that ]10158401015840(119880119899) isin Rlowast+

(ii) Let us show that WMV119880]10158401015840(119891) = WMV119880

120583 (119891) Wealready know that WMV119880

]1015840(119891) = WMV119880] (119891) =

WMV119880120583 (119891) Let 119899 isin N

1]10158401015840 (119880119899) int119880119899 119891119889 (]10158401015840)= 119905] (119880119899)(119905] + (1 minus 119905) ]1015840) (119880119899)

1119905] (119880119899) int119880119899 119891119889 (119905])+ (1 minus 119905) ]1015840 (119880119899)(119905] + (1 minus 119905) ]1015840) (119880119899)

1(1 minus 119905) ]1015840 (119880119899)sdot int

119880119899

119891119889 ((1 minus 119905) ]1015840) = 119905] (119880119899)(119905] + (1 minus 119905) ]1015840) (119880119899)+ (1 minus 119905) ]1015840 (119880119899)(119905] + (1 minus 119905) ]1015840) (119880119899)WMV119880

120583 (119891)+ 119905] (119880119899)(119905] + (1 minus 119905) ]1015840) (119880119899)

1] (119880119899)

sdot int119880119899

119891119889 (]) minusWMV119880120583 (119891)

+ (1 minus 119905) ]1015840 (119880119899)(119905] + (1 minus 119905) ]1015840) (119880119899) 1

]1015840 (119880119899) int119880119899 119891119889]1015840

minusWMV119880120583 (119891)

(43)

Now we remark that

1 = 119905] (119880119899)(119905] + (1 minus 119905) ]1015840) (119880119899) +(1 minus 119905) ]1015840 (119880119899)(119905] + (1 minus 119905) ]1015840) (119880119899) (44)


and that


119905] (119880119899)(119905] + (1 minus 119905) ]1015840) (119880119899)sdot 1

] (119880119899) int119880119899 119891119889 (]) minusWMV119880120583 (119891)

(45)

since 119905](119880119899)(119905] + (1 minus 119905)]1015840)(119880119899) isin [0 1] and lim119899rarr+infin(1](119880119899)) int119880119899 119891119889(]) =WMV119880

] (119891) =WMV119880120583 (119891) and finally that

lim119899rarr+infin(119905](119880119899)(119905] + (1 minus 119905)]1015840)(119880119899))(1](119880119899)) int119880119899 119891119889(]) minusWMV119880

120583 (119891) = 0 the same way Thus

WMV119880]10158401015840 (119891) = lim

119899rarr+infin

1]10158401015840 (119880119899) int119880119899 119891119889 (]10158401015840)

=WMV119880120583 (119891)

(46)

]10158401015840 isinM(120583 119880 119865) thusM(120583 119880 119865) is a convex cone33 Asymptotic Comparison of Radon Measures We nowturn to the numberΘ that appeared in Proposition 19 In thissection 120583 and ] are fixed Radon measures and 119880 is a fixedsequence in Ren120583 cap Ren]

Proposition 22 One has the following

(1) Θ(120583 ]) isin [0 1](2) Θ(120583 ]) = 1 minus Θ(] 120583)

Proof The proof is obvious

Definition 23 One has the following

(1) ] = 119900119880(120583) if Θ(120583 ]) = 1(2) ] = 119874119880(120583) if Θ(120583 ]) gt 0(3) ]sim119880120583 if Θ(120583 ]) = 12Let us now compare three measures 120583 ] and 120588119880 is here

a fixed arbitrary sequence

Lemma 24 Let 120583 and ] be two measures and let 119880 isin 119877119890119899120583 cap119877119890119899]Θ(120583 ]) = 11 + 120579 (120583 ]) (47)

where 120579(120583 ]) = lim119899rarr+infin(](119880119899)120583(119880119899)) isin R = [0 +infin]Proof The proof is obvious

Proposition 25 Let 119880 isin 119877119890119899120583 cap 119877119890119899] cap 119877119890119899120588(1) 120579(120583 120588) = 120579(120583 ])120579(] 120588) if (120579(120583 ]) 120579(] 120588)) notin (0 +infin)(+infin 0)(2) Θ(120583 120588) = Θ(120583 ])Θ(] 120588)(2Θ(120583 ])Θ(] 120588) minusΘ(120583 ]) minusΘ(] 120588) + 1) if (Θ(120583 ]) Θ(] 120588)) notin (1 0) (0 1)

Proof Let 119899 isin N We have 120583(119880119899)(120583 + ])(119880119899) = 1(1 +](119880119899)120583(119880119899)) ](119880119899)(] + 120588)(119880119899) = 1(1 + 120588(119880119899)](119880119899)) and120583(119880119899)(120583 + 120588)(119880119899) = 1(1 + 120588(119880119899)120583(119880119899)) For the first partof the statement

120588 (119880119899)120583 (119880119899) =120588 (119880119899)] (119880119899)

] (119880119899)120583 (119880119899) (48)

(since these numbers are positive the equality makes sense)Thus if the limits are compatible we get (1) taking the limitsof both parts Then we express each part as

120588 (119880119899)120583 (119880119899) =(120583 + 120588) (119880119899)120583 (119880119899) minus 1

120588 (119880119899)] (119880119899) = (] + 120588) (119880119899) ] (119880119899) minus 1] (119880119899)120583 (119880119899) = (120583 + ]) (119880119899) 120583 (119880119899) minus 1

(49)

and we get

120583 (119880119899)(120583 + 120588) (119880119899) =(120583 (119880119899) (120583 + ]) (119880119899)) (] (119880119899) (] + 120588) (119880119899))2 (120583 (119880119899) (120583 + ]) (119880119899)) (] (119880119899) (] + 120588) (119880119899)) minus 120583 (119880119899) (120583 + ]) (119880119899) minus ] (119880119899) (] + 120588) (119880119899) + 1 (50)

Taking the limit we get (2)

We recover by these results a straightforward extension ofthe comparison of the asymptotic behaviour of functionsThenotation chosen in Definition 23 shows this correspondenceThrough easy calculations of 120579 orΘ one can easily see that if120583 ] and ]1015840 are comparable measures

(1) (120583 sim ]) and (] sim ]1015840) rArr (120583 sim ]1015840)(2) (120583 sim ]) hArr (] sim 120583)

(3) (120583 = 119900(])) rArr (120583 = 119874(]))(4) (120583 = 119874(])) and (] = 119874(]1015840)) rArr (120583 = 119874(]1015840))(5) (120583 = 119900(])) and (] = 119900(]1015840)) rArr (120583 = 119900(]1015840))(6) (120583 = 119874(]1015840)) and (] = 119874(]1015840)) rArr (120583 + ] = 119874(]1015840))

and other easy relations can be deduced in the same spirit

34 Limits andMean Value If119883 is for example a connectedlocally compact paracompact and not compact manifold


equipped with a Radonmeasure 120583 such that 120583(119883) = +infin anyexhaustive sequence 119870 = (119870119899)119899isinN of compact subsets of 119883 issuch that 119870 isin Ren120583 In this setting it is natural to consider119883 = 119883 cupinfin the Alexandroff compactification of119883Theorem 26 Let 119891 119883 rarr R be a bounded measurablemap which extends to 119891 119883 rarr R a continuous map atinfinThen119882119872119881119870

120583 (119891) = 119891(infin) for each exhaustive sequence 119870 ofcompact subsets of 119883Proof We can assume that 119891(infin) = 0 in other words

lim119909rarrinfin

119891 (119909) = 0 (51)

The sequence (119870119888119899)119899isinN gives a basis of neighborhood of infin

thussup119909isin119870119888119899

1003816100381610038161003816119891 (119909)1003816100381610038161003816 lt 120598 forall1205981015840 gt 0 exist1198731015840 isin N forall119899 ge 1198731015840 (52)

Moreover since lim119899rarr+infin120583(119870119899) = +infin

120583 (1198701198990) lt 12059810158401015840120583 (119870119899)

forall1198990 isin N forall12059810158401015840 gt 0 exist11987310158401015840 isin N forall119899 ge 11987310158401015840 (53)

Let 120598 gt 0 Let 1205981015840 = 1205982We set 1198990 = 1198731015840 and 12059810158401015840 = 1205982 sup119883|119891|Then forall119899 ge 119873 = max(1198990 11987310158401015840)100381610038161003816100381610038161003816100381610038161003816int119870119899 119891119889120583

100381610038161003816100381610038161003816100381610038161003816 le int119870119899 10038161003816100381610038161198911003816100381610038161003816 119889120583 = int119870119899010038161003816100381610038161198911003816100381610038161003816 119889120583 + int

119870119899minus1198701198990

10038161003816100381610038161198911003816100381610038161003816 119889120583le (sup

119883

10038161003816100381610038161198911003816100381610038161003816) 120583 (1198701198990) + 1205981015840120583 (119870119899 minus 1198701198990

) (54)

The second term is bounded by 1205981015840120583(119870119899) = 120598120583(119870119899)2 andwe majorate the first term by 12059810158401015840(sup119883|119891|)120583(119870119899) = 120598120583(119870119899)2Thus100381610038161003816100381610038161003816100381610038161003816

1120583 (119870119899) int119870119899 119891119889120583100381610038161003816100381610038161003816100381610038161003816 le 120598 forall120598 gt 0 exist119873 gt 0 forall119899 ge 119873 (55)

and hence WMV119870120583 (119891) = 0

As mentioned in the Introduction we found no straight-forward Beppo-Levy type theorem for mean values The firstcounterexample we find is for119883 = R and 120583 = 120582 the Lebesguemeasure an increasing sequence of 1198711(120582)which converges to1R (uniformly on each compact subset of R) for examplethe sequence (119890minus1199092119899)119899isinNlowast Let 119870119899 = [minus119899 minus 1 119899 + 1] and119870 = (119870119899)119899isinN We have 119870 isin Ren120582 WMV119870

120582 (1R) = 1 andWMV119870

120582 (119890minus1199092119899) = 0 by Theorem 26 We can only state thefollowing theorem on uniform convergence

Lemma 27 Let 120583 be a measure on119883 and let119880 isin 119877119890119899120583 Let 1198911and 1198912 be two functions inF119880

120583 (119883 119881) where 119881 in a sclctvsLet 119901 be a norm on 119881 If there exists 120598 isin Rlowast

+ such thatsup119909isin119883119901(1198911(119909) minus 1198912(119909)) lt 120598 then

119901 (119882119872119881119880120583 (1198911)) minus 120598 le 119901 (119882119872119881119880

120583 (1198912))le 119901 (119882119872119881119880

120583 (1198911)) + 120598 (56)

Proof Let 119899 isin N and

119901( 1120583 (119880119899) int119880119899 1198912119889120583)le 1120583 (119880119899) int119880119899 119901 (1198911 minus 1198912) 119889120583+ 119901( 1120583 (119880119899) int119880119899 1198911119889120583)

le 120598 + 119901( 1120583 (119880119899) int119880119899 1198911119889120583)

(57)

We get in the same way

119901( 1120583 (119880119899) int119880119899 1198911119889120583) minus 120598 le 119901(1120583 (119880119899) int119880119899 1198912119889120583) (58)

The result is obtained by taking the limit

Theorem 28 Let (119891119899)119899isinN isin (F119880120583 )N be a sequence which

converges uniformly on119883 to a 120583-measurable map 119891Then

(1) 119891 isin F119880120583

(2) 119882119872119881119880120583 (119891) = lim119899rarr+infin119882119872119881119880

120583 (119891119899)Proof Let 119906119899 =WMV119880

120583 (119891119899)(i) Let us prove that (119906119899) has a limit 119906 isin 119881 Let 119901 be a

norm on 119881 Let 120598 isin Rlowast+ There exists 119873 isin N such that for

each (119899119898) isin N2

sup119909isin119883

119901 (119891119899 minus 119891119898) lt 120598 (59)

Thus by Lemma 27 with 1198911 = 0 and 1198912 = 119891119899 minus 119891119898119901 (119906119899 minus 119906119898) = 119901 (WMV119880

120583 (119891119899 minus 119891119898)) le 120598 (60)

Thus the sequence (119906119899) is a Cauchy sequence Since 119881 iscomplete the sequence (119906119899) has a limit 119906 isin 119881

(ii) Moreover we remember that forall120598 gt 0 forall(119899119898) isin N2

(sup119909isin119883

119901 (119891119899 minus 119891) lt 120598) and (sup119909isin119883

119901 (119891119898 minus 119891) lt 120598)997904rArr sup

119909isin119883119901 (119891119899 minus 119891119898) lt 2120598

997904rArr 119901(WMV119880120583 (119891119899 minus 119891119898)) lt 2120598

997904rArr 119901 (119906119899 minus 119906) lt 2120598

(61)


(iii) Let us prove that 119906 = lim119899rarr+infin(1120583(119880119899)) int119880119899 119891119889120583 Let(119899 119896) isin N2

119901( 1120583 (119880119899) int119880119899 119891119889120583 minus 119906)le 119901( 1120583 (119880119899) int119880119899 119891119889120583 minus

1120583 (119880119899) int119880119899 119891119896119889120583)+ 119901( 1120583 (119880119899) int119880119899 119891119896119889120583 minus 119906119896) + 119901 (119906119896 minus 119906)

(62)

Let 120598 isin Rlowast+ Let119870 such that forall119896 gt 119870 sup119909isin119883119901(119891 minus 119891119896) lt 1205988

Then

119901( 1120583 (119880119899) int119880119899 119891119889120583 minus1120583 (119880119899) int119880119899 119891119896119889120583) lt

1205988 119901 (119906119896 minus 119906) lt 1205984

(63)

Let119873 such that for each 119899 gt 119873

119901( 1120583 (119880119899) int119880119899 119891119870+1119889120583 minus 119906119870+1) lt1205988 (64)

Then by the same arguments for each 119896 gt 119870119901( 1120583 (119880119899) int119880119899 119891119896119889120583 minus 119906119896) lt

31205988 (65)

Gathering these inequalities we get

119901( 1120583 (119880119899) int119880119899 119891119889120583 minus 119906) lt1205988 + 31205988 + 1205984 lt 120598 (66)

35 Invariance of the Mean Value with respect to the LebesgueMeasure In this section 119883 = R119898 with 119899 isin Nlowast 120582 isthe Lebesgue measure 119870 = (119870119899)119899isinN is the renormalizationprocedure defined by

119870119899 = [minus119899 minus 1 119899 + 1]119899 (67)

and 119871 = (119871119899)119899isinN is the renormalization procedure defined by

119871119899 = 119909 isin R119899 119909 le 119899 + 1 (68)

where sdot is the Euclidian normWe denote by sdot infin the supnorm and 119889infin its associated distance Let V isin R119899We use theobvious notations119870+V = (119870119899 +V)119899isinN and 119871+V = (119871119899 +V)119899isinNfor the translated sequences Let (119860 119861) isin P(119883)2We denoteby 119860Δ119861 = (119860 minus 119861) cup (119861 minus 119860) the symmetric difference ofsubsets

Proposition 29 Let V isin R119898 Let 119891 isin F119870120582 (resp 119891 isin F119871

120582) bea bounded function Let 119880 isin 119877119890119899120582 and V isin R119899 If

lim119899rarr+infin

120582 (119880119899Δ119880119899 + V)120582 (119880119899) = 0 (69)

then

(1) 119891 isin F119880+V120582 (resp 119891 isin F119880+V

120582 ) and 119882119872119881119880120582 (119891) =119882119872119881119880+V

120582 (119891)(2) for 119891V 119909 997891rarr 119891(119909 minus V) we have 119891 isin F119880

120582 and119882119872119881119880120582 (119891) = 119882119872119881119880

120582 (119891V)Proof We first notice that the second item is a reformu-lation of the first item by change of variables 119909 997891rarr 119909 minusV WMV119870+V

120582 (119891) =WMV119870120582 (119891V)

Let us now prove the first item Let 119899 isin N and

1120582 (119880119899) int119880119899 119891119889120582 minus1120582 (119880119899 + V) int119880119899+V 119891119889120582 =

1120582 (119880119899)sdot int

119880119899

119891119889120582 minus 1120582 (119880119899) int119880119899+V 119891119889120582= 1120582 (119880119899) (int119880119899minus(119880119899+V) 119891119889120582 minus int(119880119899+V)minus119880119899 119891119889120582)= 1120582 (119870119899) (int119880119899Δ(119880119899+V) (1119880119899minus(119880119899+V) minus 1(119880119899+V)minus119880119899)sdot 119891119889120582)

(70)

Let119872 = supR119898(|119891|)Then1003816100381610038161003816100381610038161003816100381610038161120582 (119880119899) int119880119899 119891119889120582 minus

1120582 (119880119899 + V) int119880119899+V 119891119889120582100381610038161003816100381610038161003816100381610038161003816

le 119872120582 (119880119899Δ119880119899 + V)120582 (119880119899) (71)

Thus we get the result

Lemma 30 One has the following

lim119899rarr+infin

120582 (119870119899Δ119870119899 + V)120582 (119870119899) = 0lim

119899rarr+infin

120582 (119871119899Δ119871119899 + V)120582 (119871119899) = 0(72)

Proof We prove it for the sequence 119870 and the proof is thesame for the sequence 119871 We have 119870119899 = (119899 + 1)1198700 thus120582(119870119899) = (119899 + 1)119898120582(1198700) and

120582 (119870119899Δ119870119899 + V) = (119899 + 1)119898 120582 (1198700Δ1198700 + 1119899 + 1V) (73)

Let

119860119899 = 119909 isin R119898 | 119889infin (119909 1205971198700) lt 2 Vinfin119899 + 1 (74)

We have1198700Δ1198700 + (1(119899 + 1))V sub 119860119899 and lim119899rarr+infin120582(119860119899) = 0Thus

lim119899rarr+infin

120582 (119870119899Δ119870119899 + V)120582 (119870119899) = 0 (75)



120582) bea bounded function

(1) Then 119891 isin F119870+V120582 (resp 119891 isin F119871+V

120582 ) and119882119872119881119870120582 (119891) =119882119872119881119870+V

120582 (119891) (resp119882119872119881119871120582 (119891) = 119882119872119881119871+V

120582 (119891))(2) Let 119891V 119909 997891rarr 119891(119909 minus V)Then 119891 isin F119870

120582 (resp 119891 isin F119871120582)

and 119882119872119881119870120582 (119891) = 119882119872119881119870

120582 (119891V) (resp 119882119872119881119871120582 (119891) =119882119872119881119871

120582 (119891V))Proof The proof for119870 and 119871 is a straightforward applicationof Proposition 29 which is valid thanks to the previouslemma

36 Example The Mean Value Induced by a Smooth MorseFunction In this example 119883 is a smooth locally compactparacompact connected oriented and noncompact mani-fold of dimension 119899 ge 1 equipped with a measure 120583 inducedby a volume form 120596 and a Morse function 119865 119883 rarr R suchthat

120583 (119865 le 119886) lt +infin forall119886 isin R (76)

For the theory ofMorse functions we refer to [19] Notice thatthere exists some value119860 isin R such that 120583(119865 lt 119860) gt 0Noticethat we can have 120583(119883) isin]0 +infin]Definition 32 Let 119891 119883 rarr 119881 be a smooth function into asclctvs 119881 Let 119905 isin [119860 +infin[We define

119868119891120583 (119891 119905) = 1120583 (119865 le 119905) int119865le119905 119891 (119909) 119889120583 (119909) (77)

and if the limit exists

WMV119865120583 (119891) = lim

119905rarr+infin119868119865120583 (119891 119905) (78)

Of course this definition is the ldquocontinuumrdquo versionof the ldquosequentialrdquo Definition 17 If 119881 is metrizable forany increasing sequence (120572119899)119899isinN isin [119860 +infin[N such thatlim119899rarr+infin120583119865 le 120572119899 ge 120583(119883) setting 119880119899 = 119865 le 120572119899

WMV119880120583 (119891) =WMV119865

120583 (119891) (79)

and converselyWMF119865120583(119891) exists if WMF119880120583 (119891) exists and doesnot depend on the choice of the sequence (120572119899)119899isinN

Moreover since 119865 is a Morse function it has isolatedcritical points and changing119883 into119883 minus 119862 where 119862 is the setof critical points of 119865 for each 119905 isin [119860 +infin[

119865 = 119905 = 119865minus1 (119905) (80)

is an (119899minus1)-dimensionalmanifold (disconnected or not)Thefirst examples that we can give are definite positive quadraticforms on a vector space in which119883 is embedded

37 Application Homology as aMean Value Let119872 be a finitedimensional manifold equipped with a Riemannian metric 119892and the corresponding Laplace-Beltrami operatorΔ and withfinite dimensional de Rham cohomology space 119867lowast(119872 119877)

One of the standard results of Hodge theory is the onto andone-to-one map between 119867lowast(119872 119877) and the space of 1198712-harmonic formsHmade by integration over simplexes

119868 H 997888rarr 119867lowast (119872 119877) 120572 997891997888rarr 119868 (120572) (81)

where

119868 (120572) 119904 simplex 997891997888rarr 119868 (120572) (119904) = int119904120572 (82)

We have assumed here that the simplex has the order ofthe harmonic form This is mathematically coherent statingint119904120572 = 0 if 119904 and 120572 do not have the same order Let 120582 be the

Lebesgue measure on H with respect to the scalar productinduced by the 1198712-scalar product Let 119880 = (119880119899)119899isinN be thesequence of Euclidian balls centered at 0 such that for each119899 isin N the ball 119880119899 is of radius 119899Proposition 33 Assume that119867lowast(119872 119877) is finite dimensionalLet 119904 be a simplex Let

120593119904 = |119868 (sdot) (119904)|1 + |119868 (sdot) (119904)| (83)

The cohomology class of 119904 is null if and only if119882119872119881119880

120582 (120593119904) = 0 (84)

Proof (i) If the cohomology class of 119904 is null then forall120572 isinH int

119904120572 = 0 thus 120593119904(120572) = 0 Finally

WMV119880120582 (120593119904) = 0 (85)

(ii) If the cohomology class of 119904 is not null let 120572119904 be thecorresponding element in H We have int

119904120572119904 = 1 Let 120587119904 be

the projection onto the 1-dimensional vector space spannedby 120572119904 Let 119899 isin Nlowast Let

119881119899 = 120572 isin 119880119899 such that10038161003816100381610038161003816100381610038161003816int119904 120572

10038161003816100381610038161003816100381610038161003816 gt 12 (86)

Then

119881119899 = 119880119899 cap 120587minus1119904 ([minus1 1] sdot 120572119904) (87)

Moreover

inf120572isin119881119899

120593119904 (120572) = 12 120582 (119881119899) gt 120582 (119880119899minus1)

(88)

Then

int119880119899

120593119904119889120582 ge int119881119899

120593119904119889120582 ge 120582 (119881119899)2 (89)


Thus

WMV119880120582 (120593119904) = lim

119899rarr+infin

1120582 (119880119899) int119880119899 120593119904119889120582ge lim

119899rarr+infin

120582 (119881119899)2120582 (119880119899) ge lim119899rarr+infin

120582 (119880119899minus1)2120582 (119880119899) =12

= 0(90)

Remark 34 Avery easy application ofTheorem26 also showsthat the map 119904 997891rarrWMV119880

120582 (120593119904) is a 0 1-valued map

4 The Mean Value on Infinite Products

41 Mean Value on an Infinite Product of Measured SpacesLet Λ be an infinite (countable continuous or other) set ofindexes Let (119883120582 120583120582)120582isinΛ or for short (119883120582)Λ be a family ofmeasured spaces as before We assume that on each space119883120582 we have fixed a sequence 119880120582 isin Ren120583120582 Let 119883119888 = prod120582isinΛ119883120582

be the Cartesian product of the sequence (119883120582)ΛDefinition 35 Let 119891 isin 1198620(119883) for the product topology 119891 iscalled cylindrical if and only if there exists Λ a finite subset ofΛ and a map 119891 isin 1198620(prod120582isinΛ119883120582) such that

119891 ((119909120582)Λ) = 119891 ((119909120582)Λ) forall (119909120582)Λ isin 119883 (91)

Then we set if 119891 isin Fprod120582isinΛ119880120582otimes120582isinΛ120583120582

WMV (119891) =WMVprod120582isinΛ119880120582otimes120582isinΛ120583120582

(119891) (92)

In this case we use the notation 119891 isin F (here subsidiarynotations are omitted since the sequence of measures and thesequences of renormalization are fixed in this section)

Notice that if we have Λ sub Λ 0 with the notations usedin the definition and since 119891 is constant with respect tothe variables 119909120582 indexed by 120582 isin Λ 0 minus Λ the definition ofWMV(119891) does not depend on the choice of Λ which makesit coherent

Theorem 36 Let 119891 be a cylindrical function associated withthe finite set of indexes Λ = 1205821 120582119899 and with the function119891 isin 1198620(prod120582isinΛ119883120582)

(1) Let 120582 isin Λ Let us fix 119880120582 isin 119877119890119899120583120582 Then prod120582isinΛ119880120582 =(prod120582isinΛ(119880120582)119899)119899isinN isin 119877119890119899otimes120582isinΛ120583120582 (2) If both sides are defined for each scalar-valued map119891 = 1198911205821 otimes sdot sdot sdot otimes 119891120582119899 isin F

11988012058211205831205821

otimes sdot sdot sdot otimesF1198801205821198991205831205821

we have

119882119872119881prod120582isinΛ119880120582otimes120582isinΛ120583120582

(119891) = prod120582isinΛ

119882119872119881119880120582120583120582(119891120582) (93)

For convenience of notations we shall write WMV(119891)instead of WMVprod120582isinΛ119880120582

otimes120582isinΛ120583120582(119891) Let us now consider an arbitrary

map 119891 119883 rarr 119881 which is not cylindrical (119881 is a sclctvs)Theorem 28 gives us a way to extend the notion ofmean valueby uniform convergence of sequences of cylindrical mapsBut we shall do this not only for 119883 but also for classes offunctions defined on a class of subsets of119883These classes arethe following ones

Definition 37 LetD sub 119883The domainD is called admissibleif and only if

(⨂120582isinΛ

120583120582)((prod120582isinΛ

119880120582119899) minusDΛ119899119909) = 0forall119909 isin D forallΛ finite subset of Λ forall119899 isin N

(94)

where

DΛ119899119909 = 119906 isin prod120582isinΛ119880120582119899 | exist1199091015840isin D (forall120582 isin Λ 1199091015840120582 = 119906120582)and (forall120582 isin Λ minus Λ 1199091015840120582 = 119909120582)

(95)

Definition 38 Let D be an admissible domain A function119891 D rarr 119881 is cylindrical if its value depends only on a finitenumber of coordinates indexed by a fixed finite subset of Λ

Themean value of a cylindrical function119891 is immediatelycomputed since its trace is defined on prod120582isinΛ119880120582119899 up to asubset of measure 0

Theorem39 Let119881 be a sclctvs Let119891 Drarr 119881 be the uniformlimit of a sequence (119891119899)119899isinN of cylindrical functions on D witha mean value onDThen

(1) the sequence (119882119872119881(119891119899))119899isinN has a limit(2) this limit does not depend on the sequence (119891119899)119899isinN but

only on 119891Proof Let 119906119899 =WMV119880



each (119899119898) isin N2

sup119909isin119883

119901 (119891119899 minus 119891119898) lt 120598 (96)


120583 (119891119899 minus 119891119898)) le 120598 (97)


(ii) Now let us consider another sequence (1198911015840119899) of cylin-

drical functions which converges uniformly to 119891 In orderto finish the proof of the theorem let us prove that 119906 =lim119899rarr+infinWMV119880

120583 (1198911015840119899)


Let 119899 isin NWe define

11989110158401015840119899 =

11989110158401198992 if 119899 is even

119891(119899minus1)2 if 119899 is odd (98)

This sequence again converges uniformly to 119891 and is thus aCauchy sequence The sequence (WMV119880

120583 (11989110158401015840119899 )119899isinN has a limit1199061015840 isin 119881 Extracting the subsequences (119891119899)119899isinN = (11989110158401015840

2119899+1)119899isinN and(1198911015840119899)119899isinN = (11989110158401015840

2119899+1)119899isinN we get

1199061015840 = lim119899rarr+infin

WMV119880120583 (11989110158401015840

119899 )= lim

119899rarr+infinWMV119880

120583 (119891119899) = 119906lim


120583 (1198911015840119899) = lim


120583 (11989110158401015840119899 ) = 119906

(99)

By the way the following definition is justified

Definition 40 Let 119881 be a sclctvs Let 119891 D rarr 119881 be theuniform limit of a sequence (119891119899)119899isinN of cylindrical functionsonD with a mean value onDThen

WMV119880120583 (119891) = lim


120583 (119891119899) (100)

Trivially themapWMV119880120583 is linear as well as in the context

of Proposition 18

42 Application The Mean Value on Marked Infinite Con-figurations Let 119883 be a locally compact and paracompactmanifold orientable and let 120583 be a measure on 119883 inducedby a volume form In the following we have either of thefollowing

(i) if119883 is compact setting 1199090 isin 119883Γ = (119906119899)119899isinN isin 119883N | lim 119906119899 = 1199090 forall (119899119898) isin N

2 119899= 119898 997904rArr 119906119899 = 119906119898 (101)

(ii) if 119883 is not compact setting (119870119899)119899isinN an exhaustivesequence of compact subspaces of119883

119874Γ = (119906119899)119899isinN isin 119883N | forall119901 isin N 10038161003816100381610038161003816119906119899 119899 isin N cap 119870119901

10038161003816100381610038161003816lt +infin forall (119899119898) isin N

2 119899 = 119898 997904rArr 119906119899 = 119906119898 (102)

The first setting was first defined by Ismaginov VershikGelrsquofand and Graev see for example [20] for a recentreference and the second one has been extensively studiedby Albeverio Daletskii Kondratiev and Lytvynov see forexample [21] Alternatively Γ can be seen as a set of countablesums of Dirac measures equipped with the topology of vagueconvergence

For the following we also need the set of ordered finite119896-configurations119874Γ119896 = (1199061 119906119896) isin 119883119896 | forall (119899119898)isin N

2 (1 le 119899 lt 119898 le 119896) 997904rArr (119906119899 = 119906119898) (103)

Assume now that119883 is equipped with a Radon measure 120583 Letus fix 119880 isin Ren120583 Notice first that for each (119899 119896) isin Nlowast times N

120583otimes119899 (119880119899119896 ) = 120583otimes119899 ((1199091 119909119899)

isin 119880119899119896 | forall (119894 119895) (0 le 119894 lt 119895 le 119899) 997904rArr (119909119894 = 119909119895)) (104)

In other words the set of 119899-uples for which there exists twocoordinates that are equal is of measure 0This shows that119874Γis an admissible domain in 119883N and enables us to write for abounded cylindrical function 119891

WMV119880120583 (119891) =WMV119880Ord(119891)

120583otimesOrd(119891) (119891) (105)

since 119891 is defined up to a subset of measure 0 on each119880Ord(119891)119896 for 119896 isin N By the way Theorem 39 applies in

this setting Notice also that the normalization sequence 119880on 119874Γ is induced from the normalization sequence on 119883NThis implies heuristically that cylindrical functions with aweak mean value with respect to 119880 are in a sense smallperturbations of functions on119883NThis is why we can modifythe sequence 119880 on 119874Γ in the following way let 120593 R+ rarrRlowast

+ be a function such that lim119909rarr+infin120593 = 0 Then if 119891 is acylindrical function on 119874Γ we set

119880119899120593 = 119880119899 minus (119909119894)1le119894le119899 | exist (119894 119895) such that 119894 lt 119895and 119889 (119909119894 119909119895) lt 120593 (119899) (106)

5 Mean Value for HeuristicLebesgue Measures

Definition 41 A normalized Frechet space is a pair (119865119867)where

(1) 119865 is a Frechet space(2) 119867 is a Hilbert space(3) 119865 sub 119867(4) 119865 is dense in119867Another way to understand this definition is the follow-

ing we choose a pre-Hilbert norm on the Frechet space 119865Then119867 is the completion of 119865Definition 42 Let 119881 be a sclctvs A function 119891 119865 rarr 119881is cylindrical if there exists 119865119891 a finite dimensional affinesubspace of 119865 for which if 120587 is the orthogonal projection120587 119865 rarr 119865119891 such that

119891 (119909) = 119891 ∘ 120587 (119909) forall119909 isin 119865 (107)


Remark 43 This construction is canonical inmany examplesFor instance themap 120593119904 defined in the example of Section 37extends to a cylindrical function on 1198712-forms and hence theconstruction described above applies to this extension map

Proposition 44 Let (119891119899)119899isinN be a sequence of cylindricalfunctions There exists a unique sequence (119865119891119899)119899isinN increasingfor sub for which forall119898 isin N 119865119891119898 is the minimal affine space forwhich

119891119899 ∘ 120587119898 = 119891119899 forall119899 le 119898 (108)

Proof We construct the sequence by induction

(i) 1198651198910 is the minimal affine subspace of 119865 for whichDefinition 42 applies to 1198910

(ii) Let 119899 isin N Assume that we have constructed 119865119891119899 Let 119865 be the minimal affine subspace of 119865 for whichDefinition 42 applies to 119891119899+1We set

119865119891119899+1 = 119865119891119899 + 119865 (109)

(this is the minimal affine subspace of 119865 which contains both119865119891119899 and 119865) If and 120587119899+1 are the orthogonal projections into119865 and 119865119891119899+1 then119891119899+1 = 119891119899+1 ∘ = 119891119899+1 ∘ 120587119899+1 (110)

This ends the proof

We now develop renormalization procedures on 119865inspired from Section 41 using orthogonal projections to 1-dimensional vector subspaces In these approaches a finitedimensional Euclidian space is equipped with its Lebesguemeasure noted by 120582 in any dimension The Euclidian normsare induced by the pre-Hilbert norm on 119865 for any finitedimensional vector subspace of 11986551 Mean Value by Infinite Product Let 119891 be a boundedfunction which is the uniform limit of a sequence ofcylindrical functions (119891119899)119899isinN Here an orthonormal basis(119890119896)119896isinN is obtained by induction completing at each step anorthonormal basis of 119865119891119899 by an orthonormal basis of 119865119891119899+1 Thus we can identify 119865 with a subset D of RN which isinvariant under change of a finite number of coordinatesThisqualifies it as admissible since with the notations used inDefinition 37

(prod120582isinΛ

119880120582119899) minusDΛ119899119909 = 0 (111)

for any set of renormalization procedures inRN as defined inSection 41 so that Theorem 39 applies We note by

WMV120582 (119891) (112)

this value We remark that we already know by Theorem 39that thismean value does not depend on the sequence (119891119899)119899isinN

only once the sequence (119865119891119899)119899isinN is fixed In other words twosequences (119891119899)119899isinN and (1198911015840

119899)119899isinNwhich converge uniformly to119891a priori lead to the samemean value if 119865119891119899 = 1198651198911015840119899 (maybe up toreindexation) From heuristic calculations it seems to comefrom the choice of the renormalization procedure which isdependent on the basis chosen more than from the sequence(119865119891119899)119899isinN52 Invariance We notice three types of invariance scaleinvariance translation invariance and invariance under theorthogonal (or unitary) group

Proposition 45 Let 120572 isin Nlowast Let119891 be a function on an infinitedimensional vector space 119865with mean value Let 119891120572 119909 isin 119865 997891rarr119891(120572119909)Then 119891120572 has a mean value and

119882119872119881120582 (119891120572) = 119882119872119881120582 (119891) (113)

Proof Let (119891119899)119899isinN be a sequence which converges uniformlyto 119891Then with the notations above the sequence ((119891119899)120572)119899isinNconverges uniformly to 119891120572 Let119898 = dim(119865119891119899)

WMV120582 ((119891119899)120572) =WMV120572minus119898120582 (119891119899) =WMV120582 (119891119899) (114)

By Proposition 20 and remarking that for the fixed renormal-ization sequence above this change of variables consists inextracting a subsequence of renormalizationThus taking thelimit we get

119882119872119881120582 (119891120572) = 119882119872119881120582 (119891) (115)

Proposition46 Let V isin 119865 Let119891 be a function on119865withmeanvalue Let 119891V 119909 isin 119865 997891rarr 119891(119909 + V)Then 119891V has a mean valueand

119882119872119881120582 (119891V) = 119882119872119881120582 (119891) (116)

Proof Let (119891119899)119899isinN be a sequence which converges uniformlyto 119891 Let V119899 = 120587119899(V) isin 119865119891119899 We have (119891119899)V = (119891119899)V119899 Then

WMV120582 (119891V) = lim119899rarr+infin

WMV120582 ((119891119899)V119899)= lim

119899rarr+infinWMV120582 (119891119899) by Proposition 31

=WMV120582 (119891) (117)

Proposition 47 Let 119880119865 be the group of unitary operators of119867 which restricts to a bounded map 119865 rarr 119865 together with itsinverse Let 119906 isin 119880119865 Let 119891 be a map with mean value Then119891 ∘ 119906 has a mean value and

119882119872119881120582 (119891 ∘ 119906) = 119882119872119881120582 (119891) (118)

This last proposition becomes obvious after remarkingthat we transform the sequence (119865119891119899)119899isinN into the orthogonalsequence

(119906minus1 (119865119891119899))119899isinN = (119865119891119899∘119906)119899isinN (119)

This remark shows that we get the same mean value for 119891 ∘ 119906as for 119891 by changing the orthogonal sequence


53 Invariance by Restriction With the same notations asabove let 119866 be a vector subspace of 119865 such that

⨁119899isinN

119865119891119899 sub 119866 (120)

As a consequence if119892 is the restriction of119891 to119866 the sequence(119891119899)119899isinN of cylindrical functions on 119865 restricts to a sequence(119892119899)119899isinN of cylindrical functions on 119866 Then for uniformconvergence

lim119899rarr+infin

119892119899 = 119892 (121)

and for fixed 119899 isin N we get through restriction to 119865119891119899 WMV120582 (119892119899) =WMV120582 (119891119899) (122)

Taking the limit we get

WMV120582 (119892) =WMV120582 (119891) (123)

This shows the restriction property announced in the Intro-duction

Competing Interests

The author declares that they have no competing interests

References

[1] R L Baker ldquoLebesgue measure on Rinfinrdquo Proceedings of theAmerican Mathematical Society vol 113 no 4 pp 1023ndash10291991

[2] R L Baker ldquoLebesgue measure on Rinfin IIrdquo Proceedings of theAmerican Mathematical Society vol 132 no 9 pp 2577ndash25912004

[3] M Kac and H Steinhaus ldquoSur les fonctions independantes IVrdquoStudia Mathematica vol 7 pp 1ndash15 1938

[4] C Couruneanu Almost Periodic Functions AMS Chelsea Pub-lishing 1989

[5] A L T Paterson Amenability vol 29 ofMathematical Surveysand Monographs American Mathematical Society ProvidenceRI USA 1988

[6] S Albeverio and Z Brzezniak ldquoFeynman path integrals asinfinite-dimensional oscillatory integrals some new develop-mentsrdquo Acta Applicandae Mathematicae vol 35 no 1-2 pp 5ndash26 1994

[7] S Albeverio R Hoegh-Krohn and S MazzuchiMathematicalTheory of Feynman Path Integrals An Introduction vol 523 ofLecture Notes in Mathematics Springer Berlin Germany 2ndedition 2005

[8] S Albeverio and S Mazzucchi ldquoGeneralized Fresnel integralsrdquoBulletin des Sciences Mathematiques vol 129 no 1 pp 1ndash232005

[9] S Albeverio and S Mazzucchi ldquoGeneralized infinite-dimensional Fresnel integralsrdquo Comptes Rendus Mathematiquevol 338 no 3 pp 255ndash259 2004

[10] S Albeverio and S Mazzucchi ldquoFeynman path integrals forpolynomially growing potentialsrdquo Journal of Functional Anal-ysis vol 221 no 1 pp 83ndash121 2005

[11] M Gromov Metric Structures for Riemannian and Non-Riemannian Spaces vol 152 of Progress in MathematicsBirkhaauser Boston Mass USA 1999

[12] M Gromov and V D Milman ldquoA topological application to theisopometric inequalityrdquo The American Journal of Mathematicsvol 105 pp 843ndash854 1983

[13] V Pestov Dynamics of Infinite-Dimensional Groups TheRamsey-Dvoretzky-Milman Phenomenon vol 40 of UniversityLecture Series American Mathematical Society 2006

[14] J-P Magnot ldquoInfinite dimensional integrals beyond MonteCarlo methods yet another approach to normalized infinitedimensional integralsrdquo Journal of Physics Conference Series vol410 no 1 Article ID 012003 2013

[15] V I Bogachev Differentiable Measures and the Malliavin Cal-culus vol 164 of AMS Mathematical Surveys and Monographs2010

[16] D Elworthy andA Truman ldquoFeynmanmaps Cameron-Martinformulae and anharmonic oscillatorsrdquo Annales de lrsquoInstitutHenri Poincare Physique Theorique vol 41 no 2 pp 115ndash1421984

[17] J J Duistermaat ldquoOscillatory integrals Lagrange immersionsand unfoldings of singularitiesrdquo Communications on Pure andApplied Mathematics vol 27 pp 207ndash281 1974

[18] E M SteinHarmonic Analysis Real-variable Methods Orthog-onality and Oscillatory Integrals vol 43 of Princeton UniversityPress Monographs in Harmonic Analysis III Princeton Univer-sity Press Princeton NJ USA 1993

[19] J Milnor Morse Theory Based on Lecture Notes by M Spivakand RWells vol 51 ofAnnals of Mathematics Studies PrincetonUniversity Press Princeton NJ USA 1963

[20] O V Ismaginov Representations of Infinite Dimensional GroupsTranslations of Mathematical Monographs American Mathe-matical Society 1996

[21] S Albeverio Yu G Kondratiev E W Lytvynov and G FUs ldquoAnalysis and geometry on marked configuration spacesrdquohttpsarxivorgabsmath0608344

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of functions (119891siminfin 119892 119891 = 119874(119892) and 119891 = 119900(119892)) tomeasures As a particular case the mean value with respectto the Lebesgue measure on R appears as a linear extensionof the limit at infin of functions We know very few aboutthe behaviour of the mean value of limit of functions themean value is not continuous for vague convergence butcontinuous for uniform convergence There is certainly anintermediate kind of convergence more adapted to meanvalues to be determined We also give an application of thisnotion the homology map as a mean value of a function onthe space of harmonic forms using Hodge theory

Secondly we get to infinite products of measured spacesin Section 4 Recall that there is an induced measure on aninfinite product of measured spaces only if we have spaceswith finite measures We consider cylindrical functions anddefine very easily their mean values as mean values offunctions defined on a finite product of measured spaceThen we extend it to functions that are uniform limits ofsequences of cylindrical functions As an application we givea definition of themean value on infinite configuration spacesfor Poisson measure

Finally we get to vector subspaces of Hilbert spacesin Section 4 This is where we decide to focus on theannounced heuristic infinite dimensional Lebesgue meanwhich is not the infinite dimensional Lebesgue measuredescribed in [1 2]Themean value is developed and we studyits invariance properties It appears invariant by translationand by scaling and also by action of the unitary groupBut the last one remains dependent on the choice of theorthonormal basis used for the definition which is classicalin the procedure of approximation by cylindrical functions[15] As a concluding remark we show that this approachhas a technical difference with the approach by measureson infinite dimensional spaces We show that the meanvalue of a bounded continuous function 119891 remains the samewhile restricting to a dense vector subspace This exhibits astriking difference from for example the Wiener measureon continuous paths for which the space of 1198671 paths is ofmeasure 0With all these elements we can now explainwhereis the originality of our approachHere the total volume is notconsidered as a constant of the total space but as a scale-likeelement to compare with the integral of a function

2 The Space of Means Spanned by Sequencesof Finite Measures

Let (119883 120583) be a measured space Following [5 13] let us fix avector subspaceF sub 119871infin(119883 120583) such that 1119883 isin FAmean onF is a linearmap120601 Frarr C such that120601(1119883) = 1Alternatelyif (119883 119889) is a metric space given F sub 1198620

119887(119883) (space ofcontinuous bounded maps) a mean on F is a linear map120601 F rarr C such that 120601(1119883) = 1 These two terminologiescome from the basic example where 120583 is a Borel probabilitymeasure on a metric space (119883 119889) for which the mean of acontinuous integrable map 119891 is its expectation value

int119883119891119889120583 (3)

and can be approximated by sequences of barycenters ofDirac measures via Monte Carlo methods

LetF0 be a vector space of (bounded measurable) mapsthat contains constant maps with values in a complete locallyconvex topological vector space (clctvs) 119881 We shall call allalong this paper M(119883F0) the space of means that is thespace of linear maps 120601 F0 rarr R such that 120601(1119883) = 1Themean 120601 can be defined on another domain 119863 sup F0 but thespace F0 will serve as reference domain Moreover we setK = R or C

21 Means Spanned by Probability Measures Let 119883 be acomplete metric space and let1198620

119887(119883) be the space of boundedK-valued continuous maps on119883We note by P(119883) the spaceof Borel probability measures on119883 Let us first set 119881 = K

Definition 1 A K-probability mean is a linear map 120591 D120591 sub1198620119887(119883) rarr K which is defined as the limit of barycenters with

K-weights of a sequence of Borel probability measures on119883 that isexist (120583119899 120572119899)119899isinN isin (P (119883) times K)N forall119898 isin N

lowast 119898sum119899=0

120572119899 = 0 and forall119891 isin 1198620119887 (119883) 120591 (119891)


1sum119898119899=0 120572119899 (

119898sum119899=0

120572119899120583119899 (119891)) (4)

We note by PMK(119883) the space of K-probability meansby PMK(119883) the set of probability means 120591 such that D120591 =1198620119887(119883) and by PM

+

R(119883) the means 120591 obtained by a sequence(120572119899)119899isinN isin Rlowast+ and we set

PM+R (119883) = PMC (119883) cap PM

+

R (119883) (5)

We have a special class spanned by the Dirac measures

Definition 2 (see [14]) A K-Dirac mean is a linear map 120591 D120591 sub 1198620

119887(119883) rarr Kwhich is defined as the limit of barycenterswith K-weights of a sequence of Dirac measures on119883 119898sum119899=0

120572119899 = 0 and forall119891 isin 1198620119887 (119883) 120591 (119891)


1sum119898119899=0 120572119899 (

119898sum119899=0

120572119899120575119909119899 (119891)) exist (119909119899 120572119899)119899isinN isin (119883 times K)N forall119898 isin N

lowast(6)

Wenote by DMK(119883) DMK(119883) DM+

R(119883) DM+R(119883)

the sets of Dirac means corresponding respectively toPMK(119883) PMK(119883) PM

+

R(119883) PM+R(119883)

Proposition 3 PMK(119883) PMK(119883) DMK(119883) andDMK(119883) are K-affine spaces




(i) P(119883) sub DM+R(119883)



119899=0 119891(119909119899) = E120583(119891) = int119883 119891119889120583Thus

120583 (119891) = lim119898rarr+infin

1119899 + 1119898sum119899=0

120575119909119899 (119891) (7)


119887(119883))1015840 Since PMR(119883) sub (1198620119887(119883))1015840

we get the result




119860120576 = 119909 isin 119883 | 119889 (119860 119909) lt 120576 (8)


satisfying


120583119899 (119860119899) gt 0 (10)

and then

lim119899rarr+infin

120583119899 ((119860119899)120576) = 1 forall120576 gt 0 (11)


119883119899 sub 119883119899+1 forall119899 isin N (12)


219




119878119899 (13)


such that




119871119872X (119891) = 119886 (15)


120583119899 = 1119870119899120582 (119870119899)120582 (16)




int119883119899

119891119889120583119899 (17)

if the limit exists






LM119883 (119891) = lim119899rarr+infin

120591119899 (119891) (18)





119875119896 (1199091 1199092 ) isin RN 997891997888rarr (1199091 1199092 119909119896 0 ) (19)


such that

119891 = 119891 ∘ 119875119897 (20)



lowast (21)




lim120598rarr0

intR119899119890119894Φ(119909)119891 (119909) 120593 (120598119909) 119889119909 (22)


int119900

R119899119890119894Φ(119909)119891 (119909) 119889119909 (23)



int119900

Im119875119899

119890(1198942ℎ)|119875119899(119909)|2119891 (119875119899 (119909)) 119889 (119875119899 (119909))sdot int119900

Im119875119899

119890(1198942ℎ)|119875119899(119909)|2119889 (119875119899 (119909))minus1(24)


int119900

H

119890(1198942ℎ)|119909|2119891 (119909) 119889 (119909) (25)



119891 997891997888rarr int119900

R119899119890119894Φ(119909)119891 (119909) 119889119909 isin PMC (R119899) (26)

(ii) the map


int119900

R119899119890119894Φ(119909)119891 (119909) 119889119909 (27)






119891 = 119891 ∘ 119875 (28)

and let

119878119875 = 119878 ∘ 119875 (29)



lowast (30)


119909 = (1199091 119909119889) isin R119889 997891997888rarr 120595 (119909)

= 1radic2120587 (int1199091

minusinfin119890119905211198891199051 int119909119889

minusinfin1198901199052119889119889119905119899)

(31)

or

119909 = (1199091 119909119889) isin R119889 997891997888rarr 120595 (119909)

= (Argth (1199091) Argth (119909119889))(32)


intR119899119890119894Φ(119909)119891 (119909) 120593 (120598119909) 119889119909 = int

R119899119890119894Φ∘120595minus1(119909)119891 ∘ 120595minus1 (119909) 120593

∘ 120595minus1 (120598119909) 10038161003816100381610038161003816det (119863119909120595minus1)10038161003816100381610038161003816 119889119909(33)









WMV119880120583 (119891) = lim

119899rarr+infin

1120583 (119880119899) int119880119899 119891119889120583 (34)




120583 (119891+) +WMV119880120583 (119891minus)







MV120583 (1119883) = 1 (35)


MV120575119909(119891) = 119891 (119909) (36)


MV120583 (119891) = 1120583 (119883) int119883 119891119889120583 (37)




R119892119889120582 lt+infin so that

MV120582 (119892) = 0 (38)



119895=0[2(119899 + 119895)120587 (2(119899 + 119895) + 1)120587] ThenWMV1198801015840



WMV119880120574 (119906119899) = lim

119899rarr+infin

1119899 + 1119899sum

119896=0

119906119896 (39)

is the Cesaro limit



120583 is linear




120583+]and

119882119872119881119880120583+] (119891) = Θ (120583 ])119882119872119881119880

120583 (119891)+ Θ (] 120583)119882119872119865119880] (119891) (40)

Proof

1(120583 + ]) (119880119899) int119880119899 119891119889 (120583 + ])= 120583 (119880119899)(120583 + ]) (119880119899)

1120583 (119880119899) int119880119899 119891119889120583+ ] (119880119899)(120583 + ]) (119880119899)

1] (119880119899) int119880119899 119891119889]

(41)




119896120583 = F119880120583 and119882119872119881119880



M (120583 119880 119891)= ] | 119880 isin 119877119890119899] 119882119872119881119880

] (119891) = 119882119872119881119880120583 (119891) (42)







]1015840(119891) = WMV119880] (119891) =

WMV119880120583 (119891) Let 119899 isin N

1]10158401015840 (119880119899) int119880119899 119891119889 (]10158401015840)= 119905] (119880119899)(119905] + (1 minus 119905) ]1015840) (119880119899)

1119905] (119880119899) int119880119899 119891119889 (119905])+ (1 minus 119905) ]1015840 (119880119899)(119905] + (1 minus 119905) ]1015840) (119880119899)

1(1 minus 119905) ]1015840 (119880119899)sdot int

119880119899


120583 (119891)+ 119905] (119880119899)(119905] + (1 minus 119905) ]1015840) (119880119899)

1] (119880119899)

sdot int119880119899

119891119889 (]) minusWMV119880120583 (119891)

+ (1 minus 119905) ]1015840 (119880119899)(119905] + (1 minus 119905) ]1015840) (119880119899) 1

]1015840 (119880119899) int119880119899 119891119889]1015840

minusWMV119880120583 (119891)

(43)

Now we remark that



and that


119905] (119880119899)(119905] + (1 minus 119905) ]1015840) (119880119899)sdot 1

] (119880119899) int119880119899 119891119889 (]) minusWMV119880120583 (119891)

(45)




120583 (119891) = 0 the same way Thus

WMV119880]10158401015840 (119891) = lim

119899rarr+infin

1]10158401015840 (119880119899) int119880119899 119891119889 (]10158401015840)

=WMV119880120583 (119891)

(46)



(1) Θ(120583 ]) isin [0 1](2) Θ(120583 ]) = 1 minus Θ(] 120583)









120588 (119880119899)120583 (119880119899) =120588 (119880119899)] (119880119899)

] (119880119899)120583 (119880119899) (48)


120588 (119880119899)120583 (119880119899) =(120583 + 120588) (119880119899)120583 (119880119899) minus 1

120588 (119880119899)] (119880119899) = (] + 120588) (119880119899) ] (119880119899) minus 1] (119880119899)120583 (119880119899) = (120583 + ]) (119880119899) 120583 (119880119899) minus 1

(49)

and we get

120583 (119880119899)(120583 + 120588) (119880119899) =(120583 (119880119899) (120583 + ]) (119880119899)) (] (119880119899) (] + 120588) (119880119899))2 (120583 (119880119899) (120583 + ]) (119880119899)) (] (119880119899) (] + 120588) (119880119899)) minus 120583 (119880119899) (120583 + ]) (119880119899) minus ] (119880119899) (] + 120588) (119880119899) + 1 (50)










lim119909rarrinfin

119891 (119909) = 0 (51)


thussup119909isin119870119888119899



120583 (1198701198990) lt 12059810158401015840120583 (119870119899)



100381610038161003816100381610038161003816100381610038161003816 le int119870119899 10038161003816100381610038161198911003816100381610038161003816 119889120583 = int119870119899010038161003816100381610038161198911003816100381610038161003816 119889120583 + int

119870119899minus1198701198990

10038161003816100381610038161198911003816100381610038161003816 119889120583le (sup

119883

10038161003816100381610038161198911003816100381610038161003816) 120583 (1198701198990) + 1205981015840120583 (119870119899 minus 1198701198990

) (54)



and hence WMV119870120583 (119891) = 0


120582 (1R) = 1 andWMV119870





119901 (119882119872119881119880120583 (1198911)) minus 120598 le 119901 (119882119872119881119880

120583 (1198912))le 119901 (119882119872119881119880

120583 (1198911)) + 120598 (56)



le 120598 + 119901( 1120583 (119880119899) int119880119899 1198911119889120583)

(57)


119901( 1120583 (119880119899) int119880119899 1198911119889120583) minus 120598 le 119901(1120583 (119880119899) int119880119899 1198912119889120583) (58)




(1) 119891 isin F119880120583

(2) 119882119872119881119880120583 (119891) = lim119899rarr+infin119882119872119881119880

120583 (119891119899)Proof Let 119906119899 =WMV119880



each (119899119898) isin N2

sup119909isin119883

119901 (119891119899 minus 119891119898) lt 120598 (59)


120583 (119891119899 minus 119891119898)) le 120598 (60)



(sup119909isin119883


119901 (119891119898 minus 119891) lt 120598)997904rArr sup

119909isin119883119901 (119891119899 minus 119891119898) lt 2120598

997904rArr 119901(WMV119880120583 (119891119899 minus 119891119898)) lt 2120598

997904rArr 119901 (119906119899 minus 119906) lt 2120598

(61)





(62)


Then

119901( 1120583 (119880119899) int119880119899 119891119889120583 minus1120583 (119880119899) int119880119899 119891119896119889120583) lt

1205988 119901 (119906119896 minus 119906) lt 1205984

(63)


119901( 1120583 (119880119899) int119880119899 119891119870+1119889120583 minus 119906119870+1) lt1205988 (64)


31205988 (65)


119901( 1120583 (119880119899) int119880119899 119891119889120583 minus 119906) lt1205988 + 31205988 + 1205984 lt 120598 (66)


119870119899 = [minus119899 minus 1 119899 + 1]119899 (67)


119871119899 = 119909 isin R119899 119909 le 119899 + 1 (68)




lim119899rarr+infin

120582 (119880119899Δ119880119899 + V)120582 (119880119899) = 0 (69)

then


120582 ) and 119882119872119881119880120582 (119891) =119882119872119881119880+V


120582 and119882119872119881119880120582 (119891) = 119882119872119881119880


120582 (119891) =WMV119870120582 (119891V)


1120582 (119880119899) int119880119899 119891119889120582 minus1120582 (119880119899 + V) int119880119899+V 119891119889120582 =

1120582 (119880119899)sdot int

119880119899


(70)


1120582 (119880119899 + V) int119880119899+V 119891119889120582100381610038161003816100381610038161003816100381610038161003816

le 119872120582 (119880119899Δ119880119899 + V)120582 (119880119899) (71)



lim119899rarr+infin

120582 (119870119899Δ119870119899 + V)120582 (119870119899) = 0lim

119899rarr+infin

120582 (119871119899Δ119871119899 + V)120582 (119871119899) = 0(72)


120582 (119870119899Δ119870119899 + V) = (119899 + 1)119898 120582 (1198700Δ1198700 + 1119899 + 1V) (73)

Let



lim119899rarr+infin

120582 (119870119899Δ119870119899 + V)120582 (119870119899) = 0 (75)





120582 ) and119882119872119881119870120582 (119891) =119882119872119881119870+V

120582 (119891) (resp119882119872119881119871120582 (119891) = 119882119872119881119871+V


120582 (resp 119891 isin F119871120582)

and 119882119872119881119870120582 (119891) = 119882119872119881119870

120582 (119891V) (resp 119882119872119881119871120582 (119891) =119882119872119881119871





119868119891120583 (119891 119905) = 1120583 (119865 le 119905) int119865le119905 119891 (119909) 119889120583 (119909) (77)


WMV119865120583 (119891) = lim

119905rarr+infin119868119865120583 (119891 119905) (78)


WMV119880120583 (119891) =WMV119865

120583 (119891) (79)



119865 = 119905 = 119865minus1 (119905) (80)





where

119868 (120572) 119904 simplex 997891997888rarr 119868 (120572) (119904) = int119904120572 (82)



120593119904 = |119868 (sdot) (119904)|1 + |119868 (sdot) (119904)| (83)


120582 (120593119904) = 0 (84)


119904120572 = 0 thus 120593119904(120572) = 0 Finally

WMV119880120582 (120593119904) = 0 (85)


119904120572119904 = 1 Let 120587119904 be



10038161003816100381610038161003816100381610038161003816 gt 12 (86)

Then


Moreover

inf120572isin119881119899

120593119904 (120572) = 12 120582 (119881119899) gt 120582 (119880119899minus1)

(88)

Then

int119880119899

120593119904119889120582 ge int119881119899

120593119904119889120582 ge 120582 (119881119899)2 (89)


Thus

WMV119880120582 (120593119904) = lim

119899rarr+infin

1120582 (119880119899) int119880119899 120593119904119889120582ge lim

119899rarr+infin

120582 (119881119899)2120582 (119880119899) ge lim119899rarr+infin

120582 (119880119899minus1)2120582 (119880119899) =12

= 0(90)


120582 (120593119904) is a 0 1-valued map




119891 ((119909120582)Λ) = 119891 ((119909120582)Λ) forall (119909120582)Λ isin 119883 (91)



(119891) (92)





11988012058211205831205821


we have


(119891) = prod120582isinΛ

119882119872119881119880120582120583120582(119891120582) (93)





(⨂120582isinΛ

120583120582)((prod120582isinΛ


(94)

where


(95)








each (119899119898) isin N2

sup119909isin119883

119901 (119891119899 minus 119891119898) lt 120598 (96)


120583 (119891119899 minus 119891119898)) le 120598 (97)




120583 (1198911015840119899)



11989110158401015840119899 =

11989110158401198992 if 119899 is even

119891(119899minus1)2 if 119899 is odd (98)



2119899+1)119899isinN and(1198911015840119899)119899isinN = (11989110158401015840

2119899+1)119899isinN we get

1199061015840 = lim119899rarr+infin

WMV119880120583 (11989110158401015840

119899 )= lim


120583 (119891119899) = 119906lim


120583 (1198911015840119899) = lim


120583 (11989110158401015840119899 ) = 119906

(99)



WMV119880120583 (119891) = lim


120583 (119891119899) (100)


of Proposition 18



2 119899= 119898 997904rArr 119906119899 = 119906119898 (101)




2 119899 = 119898 997904rArr 119906119899 = 119906119898 (102)



2 (1 le 119899 lt 119898 le 119896) 997904rArr (119906119899 = 119906119898) (103)


120583otimes119899 (119880119899119896 ) = 120583otimes119899 ((1199091 119909119899)



WMV119880120583 (119891) =WMV119880Ord(119891)

120583otimesOrd(119891) (119891) (105)









119891 (119909) = 119891 ∘ 120587 (119909) forall119909 isin 119865 (107)




119891119899 ∘ 120587119898 = 119891119899 forall119899 le 119898 (108)




119865119891119899+1 = 119865119891119899 + 119865 (109)


This ends the proof


(prod120582isinΛ

119880120582119899) minusDΛ119899119909 = 0 (111)


WMV120582 (119891) (112)





119882119872119881120582 (119891120572) = 119882119872119881120582 (119891) (113)


WMV120582 ((119891119899)120572) =WMV120572minus119898120582 (119891119899) =WMV120582 (119891119899) (114)


119882119872119881120582 (119891120572) = 119882119872119881120582 (119891) (115)


119882119872119881120582 (119891V) = 119882119872119881120582 (119891) (116)



WMV120582 ((119891119899)V119899)= lim


=WMV120582 (119891) (117)


119882119872119881120582 (119891 ∘ 119906) = 119882119872119881120582 (119891) (118)






⨁119899isinN

119865119891119899 sub 119866 (120)


lim119899rarr+infin

119892119899 = 119892 (121)



WMV120582 (119892) =WMV120582 (119891) (123)


Competing Interests


References





























Volume 2014




Journal of











Journal of


Function Spaces






Algebra












(i) P(119883) sub DM+R(119883)



119899=0 119891(119909119899) = E120583(119891) = int119883 119891119889120583Thus

120583 (119891) = lim119898rarr+infin

1119899 + 1119898sum119899=0

120575119909119899 (119891) (7)


119887(119883))1015840 Since PMR(119883) sub (1198620119887(119883))1015840

we get the result




119860120576 = 119909 isin 119883 | 119889 (119860 119909) lt 120576 (8)


satisfying


120583119899 (119860119899) gt 0 (10)

and then

lim119899rarr+infin

120583119899 ((119860119899)120576) = 1 forall120576 gt 0 (11)


119883119899 sub 119883119899+1 forall119899 isin N (12)


219




119878119899 (13)


such that




119871119872X (119891) = 119886 (15)


120583119899 = 1119870119899120582 (119870119899)120582 (16)




int119883119899

119891119889120583119899 (17)

if the limit exists






LM119883 (119891) = lim119899rarr+infin

120591119899 (119891) (18)





119875119896 (1199091 1199092 ) isin RN 997891997888rarr (1199091 1199092 119909119896 0 ) (19)


such that

119891 = 119891 ∘ 119875119897 (20)



lowast (21)




lim120598rarr0

intR119899119890119894Φ(119909)119891 (119909) 120593 (120598119909) 119889119909 (22)


int119900

R119899119890119894Φ(119909)119891 (119909) 119889119909 (23)



int119900

Im119875119899

119890(1198942ℎ)|119875119899(119909)|2119891 (119875119899 (119909)) 119889 (119875119899 (119909))sdot int119900

Im119875119899

119890(1198942ℎ)|119875119899(119909)|2119889 (119875119899 (119909))minus1(24)


int119900

H

119890(1198942ℎ)|119909|2119891 (119909) 119889 (119909) (25)



119891 997891997888rarr int119900

R119899119890119894Φ(119909)119891 (119909) 119889119909 isin PMC (R119899) (26)

(ii) the map


int119900

R119899119890119894Φ(119909)119891 (119909) 119889119909 (27)






119891 = 119891 ∘ 119875 (28)

and let

119878119875 = 119878 ∘ 119875 (29)



lowast (30)


119909 = (1199091 119909119889) isin R119889 997891997888rarr 120595 (119909)

= 1radic2120587 (int1199091

minusinfin119890119905211198891199051 int119909119889

minusinfin1198901199052119889119889119905119899)

(31)

or

119909 = (1199091 119909119889) isin R119889 997891997888rarr 120595 (119909)

= (Argth (1199091) Argth (119909119889))(32)


intR119899119890119894Φ(119909)119891 (119909) 120593 (120598119909) 119889119909 = int

R119899119890119894Φ∘120595minus1(119909)119891 ∘ 120595minus1 (119909) 120593

∘ 120595minus1 (120598119909) 10038161003816100381610038161003816det (119863119909120595minus1)10038161003816100381610038161003816 119889119909(33)









WMV119880120583 (119891) = lim

119899rarr+infin

1120583 (119880119899) int119880119899 119891119889120583 (34)




120583 (119891+) +WMV119880120583 (119891minus)







MV120583 (1119883) = 1 (35)


MV120575119909(119891) = 119891 (119909) (36)


MV120583 (119891) = 1120583 (119883) int119883 119891119889120583 (37)




R119892119889120582 lt+infin so that

MV120582 (119892) = 0 (38)



119895=0[2(119899 + 119895)120587 (2(119899 + 119895) + 1)120587] ThenWMV1198801015840



WMV119880120574 (119906119899) = lim

119899rarr+infin

1119899 + 1119899sum

119896=0

119906119896 (39)

is the Cesaro limit



120583 is linear




120583+]and

119882119872119881119880120583+] (119891) = Θ (120583 ])119882119872119881119880

120583 (119891)+ Θ (] 120583)119882119872119865119880] (119891) (40)

Proof

1(120583 + ]) (119880119899) int119880119899 119891119889 (120583 + ])= 120583 (119880119899)(120583 + ]) (119880119899)

1120583 (119880119899) int119880119899 119891119889120583+ ] (119880119899)(120583 + ]) (119880119899)

1] (119880119899) int119880119899 119891119889]

(41)




119896120583 = F119880120583 and119882119872119881119880



M (120583 119880 119891)= ] | 119880 isin 119877119890119899] 119882119872119881119880

] (119891) = 119882119872119881119880120583 (119891) (42)







]1015840(119891) = WMV119880] (119891) =

WMV119880120583 (119891) Let 119899 isin N

1]10158401015840 (119880119899) int119880119899 119891119889 (]10158401015840)= 119905] (119880119899)(119905] + (1 minus 119905) ]1015840) (119880119899)

1119905] (119880119899) int119880119899 119891119889 (119905])+ (1 minus 119905) ]1015840 (119880119899)(119905] + (1 minus 119905) ]1015840) (119880119899)

1(1 minus 119905) ]1015840 (119880119899)sdot int

119880119899


120583 (119891)+ 119905] (119880119899)(119905] + (1 minus 119905) ]1015840) (119880119899)

1] (119880119899)

sdot int119880119899

119891119889 (]) minusWMV119880120583 (119891)

+ (1 minus 119905) ]1015840 (119880119899)(119905] + (1 minus 119905) ]1015840) (119880119899) 1

]1015840 (119880119899) int119880119899 119891119889]1015840

minusWMV119880120583 (119891)

(43)

Now we remark that



and that


119905] (119880119899)(119905] + (1 minus 119905) ]1015840) (119880119899)sdot 1

] (119880119899) int119880119899 119891119889 (]) minusWMV119880120583 (119891)

(45)




120583 (119891) = 0 the same way Thus

WMV119880]10158401015840 (119891) = lim

119899rarr+infin

1]10158401015840 (119880119899) int119880119899 119891119889 (]10158401015840)

=WMV119880120583 (119891)

(46)



(1) Θ(120583 ]) isin [0 1](2) Θ(120583 ]) = 1 minus Θ(] 120583)









120588 (119880119899)120583 (119880119899) =120588 (119880119899)] (119880119899)

] (119880119899)120583 (119880119899) (48)


120588 (119880119899)120583 (119880119899) =(120583 + 120588) (119880119899)120583 (119880119899) minus 1

120588 (119880119899)] (119880119899) = (] + 120588) (119880119899) ] (119880119899) minus 1] (119880119899)120583 (119880119899) = (120583 + ]) (119880119899) 120583 (119880119899) minus 1

(49)

and we get

120583 (119880119899)(120583 + 120588) (119880119899) =(120583 (119880119899) (120583 + ]) (119880119899)) (] (119880119899) (] + 120588) (119880119899))2 (120583 (119880119899) (120583 + ]) (119880119899)) (] (119880119899) (] + 120588) (119880119899)) minus 120583 (119880119899) (120583 + ]) (119880119899) minus ] (119880119899) (] + 120588) (119880119899) + 1 (50)










lim119909rarrinfin

119891 (119909) = 0 (51)


thussup119909isin119870119888119899



120583 (1198701198990) lt 12059810158401015840120583 (119870119899)



100381610038161003816100381610038161003816100381610038161003816 le int119870119899 10038161003816100381610038161198911003816100381610038161003816 119889120583 = int119870119899010038161003816100381610038161198911003816100381610038161003816 119889120583 + int

119870119899minus1198701198990

10038161003816100381610038161198911003816100381610038161003816 119889120583le (sup

119883

10038161003816100381610038161198911003816100381610038161003816) 120583 (1198701198990) + 1205981015840120583 (119870119899 minus 1198701198990

) (54)



and hence WMV119870120583 (119891) = 0


120582 (1R) = 1 andWMV119870





119901 (119882119872119881119880120583 (1198911)) minus 120598 le 119901 (119882119872119881119880

120583 (1198912))le 119901 (119882119872119881119880

120583 (1198911)) + 120598 (56)



le 120598 + 119901( 1120583 (119880119899) int119880119899 1198911119889120583)

(57)


119901( 1120583 (119880119899) int119880119899 1198911119889120583) minus 120598 le 119901(1120583 (119880119899) int119880119899 1198912119889120583) (58)




(1) 119891 isin F119880120583

(2) 119882119872119881119880120583 (119891) = lim119899rarr+infin119882119872119881119880

120583 (119891119899)Proof Let 119906119899 =WMV119880



each (119899119898) isin N2

sup119909isin119883

119901 (119891119899 minus 119891119898) lt 120598 (59)


120583 (119891119899 minus 119891119898)) le 120598 (60)



(sup119909isin119883


119901 (119891119898 minus 119891) lt 120598)997904rArr sup

119909isin119883119901 (119891119899 minus 119891119898) lt 2120598

997904rArr 119901(WMV119880120583 (119891119899 minus 119891119898)) lt 2120598

997904rArr 119901 (119906119899 minus 119906) lt 2120598

(61)





(62)


Then

119901( 1120583 (119880119899) int119880119899 119891119889120583 minus1120583 (119880119899) int119880119899 119891119896119889120583) lt

1205988 119901 (119906119896 minus 119906) lt 1205984

(63)


119901( 1120583 (119880119899) int119880119899 119891119870+1119889120583 minus 119906119870+1) lt1205988 (64)


31205988 (65)


119901( 1120583 (119880119899) int119880119899 119891119889120583 minus 119906) lt1205988 + 31205988 + 1205984 lt 120598 (66)


119870119899 = [minus119899 minus 1 119899 + 1]119899 (67)


119871119899 = 119909 isin R119899 119909 le 119899 + 1 (68)




lim119899rarr+infin

120582 (119880119899Δ119880119899 + V)120582 (119880119899) = 0 (69)

then


120582 ) and 119882119872119881119880120582 (119891) =119882119872119881119880+V


120582 and119882119872119881119880120582 (119891) = 119882119872119881119880


120582 (119891) =WMV119870120582 (119891V)


1120582 (119880119899) int119880119899 119891119889120582 minus1120582 (119880119899 + V) int119880119899+V 119891119889120582 =

1120582 (119880119899)sdot int

119880119899


(70)


1120582 (119880119899 + V) int119880119899+V 119891119889120582100381610038161003816100381610038161003816100381610038161003816

le 119872120582 (119880119899Δ119880119899 + V)120582 (119880119899) (71)



lim119899rarr+infin

120582 (119870119899Δ119870119899 + V)120582 (119870119899) = 0lim

119899rarr+infin

120582 (119871119899Δ119871119899 + V)120582 (119871119899) = 0(72)


120582 (119870119899Δ119870119899 + V) = (119899 + 1)119898 120582 (1198700Δ1198700 + 1119899 + 1V) (73)

Let



lim119899rarr+infin

120582 (119870119899Δ119870119899 + V)120582 (119870119899) = 0 (75)





120582 ) and119882119872119881119870120582 (119891) =119882119872119881119870+V

120582 (119891) (resp119882119872119881119871120582 (119891) = 119882119872119881119871+V


120582 (resp 119891 isin F119871120582)

and 119882119872119881119870120582 (119891) = 119882119872119881119870

120582 (119891V) (resp 119882119872119881119871120582 (119891) =119882119872119881119871





119868119891120583 (119891 119905) = 1120583 (119865 le 119905) int119865le119905 119891 (119909) 119889120583 (119909) (77)


WMV119865120583 (119891) = lim

119905rarr+infin119868119865120583 (119891 119905) (78)


WMV119880120583 (119891) =WMV119865

120583 (119891) (79)



119865 = 119905 = 119865minus1 (119905) (80)





where

119868 (120572) 119904 simplex 997891997888rarr 119868 (120572) (119904) = int119904120572 (82)



120593119904 = |119868 (sdot) (119904)|1 + |119868 (sdot) (119904)| (83)


120582 (120593119904) = 0 (84)


119904120572 = 0 thus 120593119904(120572) = 0 Finally

WMV119880120582 (120593119904) = 0 (85)


119904120572119904 = 1 Let 120587119904 be



10038161003816100381610038161003816100381610038161003816 gt 12 (86)

Then


Moreover

inf120572isin119881119899

120593119904 (120572) = 12 120582 (119881119899) gt 120582 (119880119899minus1)

(88)

Then

int119880119899

120593119904119889120582 ge int119881119899

120593119904119889120582 ge 120582 (119881119899)2 (89)


Thus

WMV119880120582 (120593119904) = lim

119899rarr+infin

1120582 (119880119899) int119880119899 120593119904119889120582ge lim

119899rarr+infin

120582 (119881119899)2120582 (119880119899) ge lim119899rarr+infin

120582 (119880119899minus1)2120582 (119880119899) =12

= 0(90)


120582 (120593119904) is a 0 1-valued map




119891 ((119909120582)Λ) = 119891 ((119909120582)Λ) forall (119909120582)Λ isin 119883 (91)



(119891) (92)





11988012058211205831205821


we have


(119891) = prod120582isinΛ

119882119872119881119880120582120583120582(119891120582) (93)





(⨂120582isinΛ

120583120582)((prod120582isinΛ


(94)

where


(95)








each (119899119898) isin N2

sup119909isin119883

119901 (119891119899 minus 119891119898) lt 120598 (96)


120583 (119891119899 minus 119891119898)) le 120598 (97)




120583 (1198911015840119899)



11989110158401015840119899 =

11989110158401198992 if 119899 is even

119891(119899minus1)2 if 119899 is odd (98)



2119899+1)119899isinN and(1198911015840119899)119899isinN = (11989110158401015840

2119899+1)119899isinN we get

1199061015840 = lim119899rarr+infin

WMV119880120583 (11989110158401015840

119899 )= lim


120583 (119891119899) = 119906lim


120583 (1198911015840119899) = lim


120583 (11989110158401015840119899 ) = 119906

(99)



WMV119880120583 (119891) = lim


120583 (119891119899) (100)


of Proposition 18



2 119899= 119898 997904rArr 119906119899 = 119906119898 (101)




2 119899 = 119898 997904rArr 119906119899 = 119906119898 (102)



2 (1 le 119899 lt 119898 le 119896) 997904rArr (119906119899 = 119906119898) (103)


120583otimes119899 (119880119899119896 ) = 120583otimes119899 ((1199091 119909119899)



WMV119880120583 (119891) =WMV119880Ord(119891)

120583otimesOrd(119891) (119891) (105)









119891 (119909) = 119891 ∘ 120587 (119909) forall119909 isin 119865 (107)




119891119899 ∘ 120587119898 = 119891119899 forall119899 le 119898 (108)




119865119891119899+1 = 119865119891119899 + 119865 (109)


This ends the proof


(prod120582isinΛ

119880120582119899) minusDΛ119899119909 = 0 (111)


WMV120582 (119891) (112)





119882119872119881120582 (119891120572) = 119882119872119881120582 (119891) (113)


WMV120582 ((119891119899)120572) =WMV120572minus119898120582 (119891119899) =WMV120582 (119891119899) (114)


119882119872119881120582 (119891120572) = 119882119872119881120582 (119891) (115)


119882119872119881120582 (119891V) = 119882119872119881120582 (119891) (116)



WMV120582 ((119891119899)V119899)= lim


=WMV120582 (119891) (117)


119882119872119881120582 (119891 ∘ 119906) = 119882119872119881120582 (119891) (118)






⨁119899isinN

119865119891119899 sub 119866 (120)


lim119899rarr+infin

119892119899 = 119892 (121)



WMV120582 (119892) =WMV120582 (119891) (123)


Competing Interests


References





























Volume 2014




Journal of











Journal of


Function Spaces






Algebra












119875119896 (1199091 1199092 ) isin RN 997891997888rarr (1199091 1199092 119909119896 0 ) (19)


such that

119891 = 119891 ∘ 119875119897 (20)



lowast (21)




lim120598rarr0

intR119899119890119894Φ(119909)119891 (119909) 120593 (120598119909) 119889119909 (22)


int119900

R119899119890119894Φ(119909)119891 (119909) 119889119909 (23)



int119900

Im119875119899

119890(1198942ℎ)|119875119899(119909)|2119891 (119875119899 (119909)) 119889 (119875119899 (119909))sdot int119900

Im119875119899

119890(1198942ℎ)|119875119899(119909)|2119889 (119875119899 (119909))minus1(24)


int119900

H

119890(1198942ℎ)|119909|2119891 (119909) 119889 (119909) (25)



119891 997891997888rarr int119900

R119899119890119894Φ(119909)119891 (119909) 119889119909 isin PMC (R119899) (26)

(ii) the map


int119900

R119899119890119894Φ(119909)119891 (119909) 119889119909 (27)






119891 = 119891 ∘ 119875 (28)

and let

119878119875 = 119878 ∘ 119875 (29)



lowast (30)


119909 = (1199091 119909119889) isin R119889 997891997888rarr 120595 (119909)

= 1radic2120587 (int1199091

minusinfin119890119905211198891199051 int119909119889

minusinfin1198901199052119889119889119905119899)

(31)

or

119909 = (1199091 119909119889) isin R119889 997891997888rarr 120595 (119909)

= (Argth (1199091) Argth (119909119889))(32)


intR119899119890119894Φ(119909)119891 (119909) 120593 (120598119909) 119889119909 = int

R119899119890119894Φ∘120595minus1(119909)119891 ∘ 120595minus1 (119909) 120593

∘ 120595minus1 (120598119909) 10038161003816100381610038161003816det (119863119909120595minus1)10038161003816100381610038161003816 119889119909(33)









WMV119880120583 (119891) = lim

119899rarr+infin

1120583 (119880119899) int119880119899 119891119889120583 (34)




120583 (119891+) +WMV119880120583 (119891minus)







MV120583 (1119883) = 1 (35)


MV120575119909(119891) = 119891 (119909) (36)


MV120583 (119891) = 1120583 (119883) int119883 119891119889120583 (37)




R119892119889120582 lt+infin so that

MV120582 (119892) = 0 (38)



119895=0[2(119899 + 119895)120587 (2(119899 + 119895) + 1)120587] ThenWMV1198801015840



WMV119880120574 (119906119899) = lim

119899rarr+infin

1119899 + 1119899sum

119896=0

119906119896 (39)

is the Cesaro limit



120583 is linear




120583+]and

119882119872119881119880120583+] (119891) = Θ (120583 ])119882119872119881119880

120583 (119891)+ Θ (] 120583)119882119872119865119880] (119891) (40)

Proof

1(120583 + ]) (119880119899) int119880119899 119891119889 (120583 + ])= 120583 (119880119899)(120583 + ]) (119880119899)

1120583 (119880119899) int119880119899 119891119889120583+ ] (119880119899)(120583 + ]) (119880119899)

1] (119880119899) int119880119899 119891119889]

(41)




119896120583 = F119880120583 and119882119872119881119880



M (120583 119880 119891)= ] | 119880 isin 119877119890119899] 119882119872119881119880

] (119891) = 119882119872119881119880120583 (119891) (42)







]1015840(119891) = WMV119880] (119891) =

WMV119880120583 (119891) Let 119899 isin N

1]10158401015840 (119880119899) int119880119899 119891119889 (]10158401015840)= 119905] (119880119899)(119905] + (1 minus 119905) ]1015840) (119880119899)

1119905] (119880119899) int119880119899 119891119889 (119905])+ (1 minus 119905) ]1015840 (119880119899)(119905] + (1 minus 119905) ]1015840) (119880119899)

1(1 minus 119905) ]1015840 (119880119899)sdot int

119880119899


120583 (119891)+ 119905] (119880119899)(119905] + (1 minus 119905) ]1015840) (119880119899)

1] (119880119899)

sdot int119880119899

119891119889 (]) minusWMV119880120583 (119891)

+ (1 minus 119905) ]1015840 (119880119899)(119905] + (1 minus 119905) ]1015840) (119880119899) 1

]1015840 (119880119899) int119880119899 119891119889]1015840

minusWMV119880120583 (119891)

(43)

Now we remark that



and that


119905] (119880119899)(119905] + (1 minus 119905) ]1015840) (119880119899)sdot 1

] (119880119899) int119880119899 119891119889 (]) minusWMV119880120583 (119891)

(45)




120583 (119891) = 0 the same way Thus

WMV119880]10158401015840 (119891) = lim

119899rarr+infin

1]10158401015840 (119880119899) int119880119899 119891119889 (]10158401015840)

=WMV119880120583 (119891)

(46)



(1) Θ(120583 ]) isin [0 1](2) Θ(120583 ]) = 1 minus Θ(] 120583)









120588 (119880119899)120583 (119880119899) =120588 (119880119899)] (119880119899)

] (119880119899)120583 (119880119899) (48)


120588 (119880119899)120583 (119880119899) =(120583 + 120588) (119880119899)120583 (119880119899) minus 1

120588 (119880119899)] (119880119899) = (] + 120588) (119880119899) ] (119880119899) minus 1] (119880119899)120583 (119880119899) = (120583 + ]) (119880119899) 120583 (119880119899) minus 1

(49)

and we get

120583 (119880119899)(120583 + 120588) (119880119899) =(120583 (119880119899) (120583 + ]) (119880119899)) (] (119880119899) (] + 120588) (119880119899))2 (120583 (119880119899) (120583 + ]) (119880119899)) (] (119880119899) (] + 120588) (119880119899)) minus 120583 (119880119899) (120583 + ]) (119880119899) minus ] (119880119899) (] + 120588) (119880119899) + 1 (50)










lim119909rarrinfin

119891 (119909) = 0 (51)


thussup119909isin119870119888119899



120583 (1198701198990) lt 12059810158401015840120583 (119870119899)



100381610038161003816100381610038161003816100381610038161003816 le int119870119899 10038161003816100381610038161198911003816100381610038161003816 119889120583 = int119870119899010038161003816100381610038161198911003816100381610038161003816 119889120583 + int

119870119899minus1198701198990

10038161003816100381610038161198911003816100381610038161003816 119889120583le (sup

119883

10038161003816100381610038161198911003816100381610038161003816) 120583 (1198701198990) + 1205981015840120583 (119870119899 minus 1198701198990

) (54)



and hence WMV119870120583 (119891) = 0


120582 (1R) = 1 andWMV119870





119901 (119882119872119881119880120583 (1198911)) minus 120598 le 119901 (119882119872119881119880

120583 (1198912))le 119901 (119882119872119881119880

120583 (1198911)) + 120598 (56)



le 120598 + 119901( 1120583 (119880119899) int119880119899 1198911119889120583)

(57)


119901( 1120583 (119880119899) int119880119899 1198911119889120583) minus 120598 le 119901(1120583 (119880119899) int119880119899 1198912119889120583) (58)




(1) 119891 isin F119880120583

(2) 119882119872119881119880120583 (119891) = lim119899rarr+infin119882119872119881119880

120583 (119891119899)Proof Let 119906119899 =WMV119880



each (119899119898) isin N2

sup119909isin119883

119901 (119891119899 minus 119891119898) lt 120598 (59)


120583 (119891119899 minus 119891119898)) le 120598 (60)



(sup119909isin119883


119901 (119891119898 minus 119891) lt 120598)997904rArr sup

119909isin119883119901 (119891119899 minus 119891119898) lt 2120598

997904rArr 119901(WMV119880120583 (119891119899 minus 119891119898)) lt 2120598

997904rArr 119901 (119906119899 minus 119906) lt 2120598

(61)





(62)


Then

119901( 1120583 (119880119899) int119880119899 119891119889120583 minus1120583 (119880119899) int119880119899 119891119896119889120583) lt

1205988 119901 (119906119896 minus 119906) lt 1205984

(63)


119901( 1120583 (119880119899) int119880119899 119891119870+1119889120583 minus 119906119870+1) lt1205988 (64)


31205988 (65)


119901( 1120583 (119880119899) int119880119899 119891119889120583 minus 119906) lt1205988 + 31205988 + 1205984 lt 120598 (66)


119870119899 = [minus119899 minus 1 119899 + 1]119899 (67)


119871119899 = 119909 isin R119899 119909 le 119899 + 1 (68)




lim119899rarr+infin

120582 (119880119899Δ119880119899 + V)120582 (119880119899) = 0 (69)

then


120582 ) and 119882119872119881119880120582 (119891) =119882119872119881119880+V


120582 and119882119872119881119880120582 (119891) = 119882119872119881119880


120582 (119891) =WMV119870120582 (119891V)


1120582 (119880119899) int119880119899 119891119889120582 minus1120582 (119880119899 + V) int119880119899+V 119891119889120582 =

1120582 (119880119899)sdot int

119880119899


(70)


1120582 (119880119899 + V) int119880119899+V 119891119889120582100381610038161003816100381610038161003816100381610038161003816

le 119872120582 (119880119899Δ119880119899 + V)120582 (119880119899) (71)



lim119899rarr+infin

120582 (119870119899Δ119870119899 + V)120582 (119870119899) = 0lim

119899rarr+infin

120582 (119871119899Δ119871119899 + V)120582 (119871119899) = 0(72)


120582 (119870119899Δ119870119899 + V) = (119899 + 1)119898 120582 (1198700Δ1198700 + 1119899 + 1V) (73)

Let



lim119899rarr+infin

120582 (119870119899Δ119870119899 + V)120582 (119870119899) = 0 (75)





120582 ) and119882119872119881119870120582 (119891) =119882119872119881119870+V

120582 (119891) (resp119882119872119881119871120582 (119891) = 119882119872119881119871+V


120582 (resp 119891 isin F119871120582)

and 119882119872119881119870120582 (119891) = 119882119872119881119870

120582 (119891V) (resp 119882119872119881119871120582 (119891) =119882119872119881119871





119868119891120583 (119891 119905) = 1120583 (119865 le 119905) int119865le119905 119891 (119909) 119889120583 (119909) (77)


WMV119865120583 (119891) = lim

119905rarr+infin119868119865120583 (119891 119905) (78)


WMV119880120583 (119891) =WMV119865

120583 (119891) (79)



119865 = 119905 = 119865minus1 (119905) (80)





where

119868 (120572) 119904 simplex 997891997888rarr 119868 (120572) (119904) = int119904120572 (82)



120593119904 = |119868 (sdot) (119904)|1 + |119868 (sdot) (119904)| (83)


120582 (120593119904) = 0 (84)


119904120572 = 0 thus 120593119904(120572) = 0 Finally

WMV119880120582 (120593119904) = 0 (85)


119904120572119904 = 1 Let 120587119904 be



10038161003816100381610038161003816100381610038161003816 gt 12 (86)

Then


Moreover

inf120572isin119881119899

120593119904 (120572) = 12 120582 (119881119899) gt 120582 (119880119899minus1)

(88)

Then

int119880119899

120593119904119889120582 ge int119881119899

120593119904119889120582 ge 120582 (119881119899)2 (89)


Thus

WMV119880120582 (120593119904) = lim

119899rarr+infin

1120582 (119880119899) int119880119899 120593119904119889120582ge lim

119899rarr+infin

120582 (119881119899)2120582 (119880119899) ge lim119899rarr+infin

120582 (119880119899minus1)2120582 (119880119899) =12

= 0(90)


120582 (120593119904) is a 0 1-valued map




119891 ((119909120582)Λ) = 119891 ((119909120582)Λ) forall (119909120582)Λ isin 119883 (91)



(119891) (92)





11988012058211205831205821


we have


(119891) = prod120582isinΛ

119882119872119881119880120582120583120582(119891120582) (93)





(⨂120582isinΛ

120583120582)((prod120582isinΛ


(94)

where


(95)








each (119899119898) isin N2

sup119909isin119883

119901 (119891119899 minus 119891119898) lt 120598 (96)


120583 (119891119899 minus 119891119898)) le 120598 (97)




120583 (1198911015840119899)



11989110158401015840119899 =

11989110158401198992 if 119899 is even

119891(119899minus1)2 if 119899 is odd (98)



2119899+1)119899isinN and(1198911015840119899)119899isinN = (11989110158401015840

2119899+1)119899isinN we get

1199061015840 = lim119899rarr+infin

WMV119880120583 (11989110158401015840

119899 )= lim


120583 (119891119899) = 119906lim


120583 (1198911015840119899) = lim


120583 (11989110158401015840119899 ) = 119906

(99)



WMV119880120583 (119891) = lim


120583 (119891119899) (100)


of Proposition 18



2 119899= 119898 997904rArr 119906119899 = 119906119898 (101)




2 119899 = 119898 997904rArr 119906119899 = 119906119898 (102)



2 (1 le 119899 lt 119898 le 119896) 997904rArr (119906119899 = 119906119898) (103)


120583otimes119899 (119880119899119896 ) = 120583otimes119899 ((1199091 119909119899)



WMV119880120583 (119891) =WMV119880Ord(119891)

120583otimesOrd(119891) (119891) (105)









119891 (119909) = 119891 ∘ 120587 (119909) forall119909 isin 119865 (107)




119891119899 ∘ 120587119898 = 119891119899 forall119899 le 119898 (108)




119865119891119899+1 = 119865119891119899 + 119865 (109)


This ends the proof


(prod120582isinΛ

119880120582119899) minusDΛ119899119909 = 0 (111)


WMV120582 (119891) (112)





119882119872119881120582 (119891120572) = 119882119872119881120582 (119891) (113)


WMV120582 ((119891119899)120572) =WMV120572minus119898120582 (119891119899) =WMV120582 (119891119899) (114)


119882119872119881120582 (119891120572) = 119882119872119881120582 (119891) (115)


119882119872119881120582 (119891V) = 119882119872119881120582 (119891) (116)



WMV120582 ((119891119899)V119899)= lim


=WMV120582 (119891) (117)


119882119872119881120582 (119891 ∘ 119906) = 119882119872119881120582 (119891) (118)






⨁119899isinN

119865119891119899 sub 119866 (120)


lim119899rarr+infin

119892119899 = 119892 (121)



WMV120582 (119892) =WMV120582 (119891) (123)


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References





























Volume 2014




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Journal of


Function Spaces






Algebra











119891 = 119891 ∘ 119875 (28)

and let

119878119875 = 119878 ∘ 119875 (29)



lowast (30)


119909 = (1199091 119909119889) isin R119889 997891997888rarr 120595 (119909)

= 1radic2120587 (int1199091

minusinfin119890119905211198891199051 int119909119889

minusinfin1198901199052119889119889119905119899)

(31)

or

119909 = (1199091 119909119889) isin R119889 997891997888rarr 120595 (119909)

= (Argth (1199091) Argth (119909119889))(32)


intR119899119890119894Φ(119909)119891 (119909) 120593 (120598119909) 119889119909 = int

R119899119890119894Φ∘120595minus1(119909)119891 ∘ 120595minus1 (119909) 120593

∘ 120595minus1 (120598119909) 10038161003816100381610038161003816det (119863119909120595minus1)10038161003816100381610038161003816 119889119909(33)









WMV119880120583 (119891) = lim

119899rarr+infin

1120583 (119880119899) int119880119899 119891119889120583 (34)




120583 (119891+) +WMV119880120583 (119891minus)







MV120583 (1119883) = 1 (35)


MV120575119909(119891) = 119891 (119909) (36)


MV120583 (119891) = 1120583 (119883) int119883 119891119889120583 (37)




R119892119889120582 lt+infin so that

MV120582 (119892) = 0 (38)



119895=0[2(119899 + 119895)120587 (2(119899 + 119895) + 1)120587] ThenWMV1198801015840



WMV119880120574 (119906119899) = lim

119899rarr+infin

1119899 + 1119899sum

119896=0

119906119896 (39)

is the Cesaro limit



120583 is linear




120583+]and

119882119872119881119880120583+] (119891) = Θ (120583 ])119882119872119881119880

120583 (119891)+ Θ (] 120583)119882119872119865119880] (119891) (40)

Proof

1(120583 + ]) (119880119899) int119880119899 119891119889 (120583 + ])= 120583 (119880119899)(120583 + ]) (119880119899)

1120583 (119880119899) int119880119899 119891119889120583+ ] (119880119899)(120583 + ]) (119880119899)

1] (119880119899) int119880119899 119891119889]

(41)




119896120583 = F119880120583 and119882119872119881119880



M (120583 119880 119891)= ] | 119880 isin 119877119890119899] 119882119872119881119880

] (119891) = 119882119872119881119880120583 (119891) (42)







]1015840(119891) = WMV119880] (119891) =

WMV119880120583 (119891) Let 119899 isin N

1]10158401015840 (119880119899) int119880119899 119891119889 (]10158401015840)= 119905] (119880119899)(119905] + (1 minus 119905) ]1015840) (119880119899)

1119905] (119880119899) int119880119899 119891119889 (119905])+ (1 minus 119905) ]1015840 (119880119899)(119905] + (1 minus 119905) ]1015840) (119880119899)

1(1 minus 119905) ]1015840 (119880119899)sdot int

119880119899


120583 (119891)+ 119905] (119880119899)(119905] + (1 minus 119905) ]1015840) (119880119899)

1] (119880119899)

sdot int119880119899

119891119889 (]) minusWMV119880120583 (119891)

+ (1 minus 119905) ]1015840 (119880119899)(119905] + (1 minus 119905) ]1015840) (119880119899) 1

]1015840 (119880119899) int119880119899 119891119889]1015840

minusWMV119880120583 (119891)

(43)

Now we remark that



and that


119905] (119880119899)(119905] + (1 minus 119905) ]1015840) (119880119899)sdot 1

] (119880119899) int119880119899 119891119889 (]) minusWMV119880120583 (119891)

(45)




120583 (119891) = 0 the same way Thus

WMV119880]10158401015840 (119891) = lim

119899rarr+infin

1]10158401015840 (119880119899) int119880119899 119891119889 (]10158401015840)

=WMV119880120583 (119891)

(46)



(1) Θ(120583 ]) isin [0 1](2) Θ(120583 ]) = 1 minus Θ(] 120583)









120588 (119880119899)120583 (119880119899) =120588 (119880119899)] (119880119899)

] (119880119899)120583 (119880119899) (48)


120588 (119880119899)120583 (119880119899) =(120583 + 120588) (119880119899)120583 (119880119899) minus 1

120588 (119880119899)] (119880119899) = (] + 120588) (119880119899) ] (119880119899) minus 1] (119880119899)120583 (119880119899) = (120583 + ]) (119880119899) 120583 (119880119899) minus 1

(49)

and we get

120583 (119880119899)(120583 + 120588) (119880119899) =(120583 (119880119899) (120583 + ]) (119880119899)) (] (119880119899) (] + 120588) (119880119899))2 (120583 (119880119899) (120583 + ]) (119880119899)) (] (119880119899) (] + 120588) (119880119899)) minus 120583 (119880119899) (120583 + ]) (119880119899) minus ] (119880119899) (] + 120588) (119880119899) + 1 (50)










lim119909rarrinfin

119891 (119909) = 0 (51)


thussup119909isin119870119888119899



120583 (1198701198990) lt 12059810158401015840120583 (119870119899)



100381610038161003816100381610038161003816100381610038161003816 le int119870119899 10038161003816100381610038161198911003816100381610038161003816 119889120583 = int119870119899010038161003816100381610038161198911003816100381610038161003816 119889120583 + int

119870119899minus1198701198990

10038161003816100381610038161198911003816100381610038161003816 119889120583le (sup

119883

10038161003816100381610038161198911003816100381610038161003816) 120583 (1198701198990) + 1205981015840120583 (119870119899 minus 1198701198990

) (54)



and hence WMV119870120583 (119891) = 0


120582 (1R) = 1 andWMV119870





119901 (119882119872119881119880120583 (1198911)) minus 120598 le 119901 (119882119872119881119880

120583 (1198912))le 119901 (119882119872119881119880

120583 (1198911)) + 120598 (56)



le 120598 + 119901( 1120583 (119880119899) int119880119899 1198911119889120583)

(57)


119901( 1120583 (119880119899) int119880119899 1198911119889120583) minus 120598 le 119901(1120583 (119880119899) int119880119899 1198912119889120583) (58)




(1) 119891 isin F119880120583

(2) 119882119872119881119880120583 (119891) = lim119899rarr+infin119882119872119881119880

120583 (119891119899)Proof Let 119906119899 =WMV119880



each (119899119898) isin N2

sup119909isin119883

119901 (119891119899 minus 119891119898) lt 120598 (59)


120583 (119891119899 minus 119891119898)) le 120598 (60)



(sup119909isin119883


119901 (119891119898 minus 119891) lt 120598)997904rArr sup

119909isin119883119901 (119891119899 minus 119891119898) lt 2120598

997904rArr 119901(WMV119880120583 (119891119899 minus 119891119898)) lt 2120598

997904rArr 119901 (119906119899 minus 119906) lt 2120598

(61)





(62)


Then

119901( 1120583 (119880119899) int119880119899 119891119889120583 minus1120583 (119880119899) int119880119899 119891119896119889120583) lt

1205988 119901 (119906119896 minus 119906) lt 1205984

(63)


119901( 1120583 (119880119899) int119880119899 119891119870+1119889120583 minus 119906119870+1) lt1205988 (64)


31205988 (65)


119901( 1120583 (119880119899) int119880119899 119891119889120583 minus 119906) lt1205988 + 31205988 + 1205984 lt 120598 (66)


119870119899 = [minus119899 minus 1 119899 + 1]119899 (67)


119871119899 = 119909 isin R119899 119909 le 119899 + 1 (68)




lim119899rarr+infin

120582 (119880119899Δ119880119899 + V)120582 (119880119899) = 0 (69)

then


120582 ) and 119882119872119881119880120582 (119891) =119882119872119881119880+V


120582 and119882119872119881119880120582 (119891) = 119882119872119881119880


120582 (119891) =WMV119870120582 (119891V)


1120582 (119880119899) int119880119899 119891119889120582 minus1120582 (119880119899 + V) int119880119899+V 119891119889120582 =

1120582 (119880119899)sdot int

119880119899


(70)


1120582 (119880119899 + V) int119880119899+V 119891119889120582100381610038161003816100381610038161003816100381610038161003816

le 119872120582 (119880119899Δ119880119899 + V)120582 (119880119899) (71)



lim119899rarr+infin

120582 (119870119899Δ119870119899 + V)120582 (119870119899) = 0lim

119899rarr+infin

120582 (119871119899Δ119871119899 + V)120582 (119871119899) = 0(72)


120582 (119870119899Δ119870119899 + V) = (119899 + 1)119898 120582 (1198700Δ1198700 + 1119899 + 1V) (73)

Let



lim119899rarr+infin

120582 (119870119899Δ119870119899 + V)120582 (119870119899) = 0 (75)





120582 ) and119882119872119881119870120582 (119891) =119882119872119881119870+V

120582 (119891) (resp119882119872119881119871120582 (119891) = 119882119872119881119871+V


120582 (resp 119891 isin F119871120582)

and 119882119872119881119870120582 (119891) = 119882119872119881119870

120582 (119891V) (resp 119882119872119881119871120582 (119891) =119882119872119881119871





119868119891120583 (119891 119905) = 1120583 (119865 le 119905) int119865le119905 119891 (119909) 119889120583 (119909) (77)


WMV119865120583 (119891) = lim

119905rarr+infin119868119865120583 (119891 119905) (78)


WMV119880120583 (119891) =WMV119865

120583 (119891) (79)



119865 = 119905 = 119865minus1 (119905) (80)





where

119868 (120572) 119904 simplex 997891997888rarr 119868 (120572) (119904) = int119904120572 (82)



120593119904 = |119868 (sdot) (119904)|1 + |119868 (sdot) (119904)| (83)


120582 (120593119904) = 0 (84)


119904120572 = 0 thus 120593119904(120572) = 0 Finally

WMV119880120582 (120593119904) = 0 (85)


119904120572119904 = 1 Let 120587119904 be



10038161003816100381610038161003816100381610038161003816 gt 12 (86)

Then


Moreover

inf120572isin119881119899

120593119904 (120572) = 12 120582 (119881119899) gt 120582 (119880119899minus1)

(88)

Then

int119880119899

120593119904119889120582 ge int119881119899

120593119904119889120582 ge 120582 (119881119899)2 (89)


Thus

WMV119880120582 (120593119904) = lim

119899rarr+infin

1120582 (119880119899) int119880119899 120593119904119889120582ge lim

119899rarr+infin

120582 (119881119899)2120582 (119880119899) ge lim119899rarr+infin

120582 (119880119899minus1)2120582 (119880119899) =12

= 0(90)


120582 (120593119904) is a 0 1-valued map




119891 ((119909120582)Λ) = 119891 ((119909120582)Λ) forall (119909120582)Λ isin 119883 (91)



(119891) (92)





11988012058211205831205821


we have


(119891) = prod120582isinΛ

119882119872119881119880120582120583120582(119891120582) (93)





(⨂120582isinΛ

120583120582)((prod120582isinΛ


(94)

where


(95)








each (119899119898) isin N2

sup119909isin119883

119901 (119891119899 minus 119891119898) lt 120598 (96)


120583 (119891119899 minus 119891119898)) le 120598 (97)




120583 (1198911015840119899)



11989110158401015840119899 =

11989110158401198992 if 119899 is even

119891(119899minus1)2 if 119899 is odd (98)



2119899+1)119899isinN and(1198911015840119899)119899isinN = (11989110158401015840

2119899+1)119899isinN we get

1199061015840 = lim119899rarr+infin

WMV119880120583 (11989110158401015840

119899 )= lim


120583 (119891119899) = 119906lim


120583 (1198911015840119899) = lim


120583 (11989110158401015840119899 ) = 119906

(99)



WMV119880120583 (119891) = lim


120583 (119891119899) (100)


of Proposition 18



2 119899= 119898 997904rArr 119906119899 = 119906119898 (101)




2 119899 = 119898 997904rArr 119906119899 = 119906119898 (102)



2 (1 le 119899 lt 119898 le 119896) 997904rArr (119906119899 = 119906119898) (103)


120583otimes119899 (119880119899119896 ) = 120583otimes119899 ((1199091 119909119899)



WMV119880120583 (119891) =WMV119880Ord(119891)

120583otimesOrd(119891) (119891) (105)









119891 (119909) = 119891 ∘ 120587 (119909) forall119909 isin 119865 (107)




119891119899 ∘ 120587119898 = 119891119899 forall119899 le 119898 (108)




119865119891119899+1 = 119865119891119899 + 119865 (109)


This ends the proof


(prod120582isinΛ

119880120582119899) minusDΛ119899119909 = 0 (111)


WMV120582 (119891) (112)





119882119872119881120582 (119891120572) = 119882119872119881120582 (119891) (113)


WMV120582 ((119891119899)120572) =WMV120572minus119898120582 (119891119899) =WMV120582 (119891119899) (114)


119882119872119881120582 (119891120572) = 119882119872119881120582 (119891) (115)


119882119872119881120582 (119891V) = 119882119872119881120582 (119891) (116)



WMV120582 ((119891119899)V119899)= lim


=WMV120582 (119891) (117)


119882119872119881120582 (119891 ∘ 119906) = 119882119872119881120582 (119891) (118)






⨁119899isinN

119865119891119899 sub 119866 (120)


lim119899rarr+infin

119892119899 = 119892 (121)



WMV120582 (119892) =WMV120582 (119891) (123)


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References





























Volume 2014




Journal of











Journal of


Function Spaces






Algebra











R119892119889120582 lt+infin so that

MV120582 (119892) = 0 (38)



119895=0[2(119899 + 119895)120587 (2(119899 + 119895) + 1)120587] ThenWMV1198801015840



WMV119880120574 (119906119899) = lim

119899rarr+infin

1119899 + 1119899sum

119896=0

119906119896 (39)

is the Cesaro limit



120583 is linear




120583+]and

119882119872119881119880120583+] (119891) = Θ (120583 ])119882119872119881119880

120583 (119891)+ Θ (] 120583)119882119872119865119880] (119891) (40)

Proof

1(120583 + ]) (119880119899) int119880119899 119891119889 (120583 + ])= 120583 (119880119899)(120583 + ]) (119880119899)

1120583 (119880119899) int119880119899 119891119889120583+ ] (119880119899)(120583 + ]) (119880119899)

1] (119880119899) int119880119899 119891119889]

(41)




119896120583 = F119880120583 and119882119872119881119880



M (120583 119880 119891)= ] | 119880 isin 119877119890119899] 119882119872119881119880

] (119891) = 119882119872119881119880120583 (119891) (42)







]1015840(119891) = WMV119880] (119891) =

WMV119880120583 (119891) Let 119899 isin N

1]10158401015840 (119880119899) int119880119899 119891119889 (]10158401015840)= 119905] (119880119899)(119905] + (1 minus 119905) ]1015840) (119880119899)

1119905] (119880119899) int119880119899 119891119889 (119905])+ (1 minus 119905) ]1015840 (119880119899)(119905] + (1 minus 119905) ]1015840) (119880119899)

1(1 minus 119905) ]1015840 (119880119899)sdot int

119880119899


120583 (119891)+ 119905] (119880119899)(119905] + (1 minus 119905) ]1015840) (119880119899)

1] (119880119899)

sdot int119880119899

119891119889 (]) minusWMV119880120583 (119891)

+ (1 minus 119905) ]1015840 (119880119899)(119905] + (1 minus 119905) ]1015840) (119880119899) 1

]1015840 (119880119899) int119880119899 119891119889]1015840

minusWMV119880120583 (119891)

(43)

Now we remark that



and that


119905] (119880119899)(119905] + (1 minus 119905) ]1015840) (119880119899)sdot 1

] (119880119899) int119880119899 119891119889 (]) minusWMV119880120583 (119891)

(45)




120583 (119891) = 0 the same way Thus

WMV119880]10158401015840 (119891) = lim

119899rarr+infin

1]10158401015840 (119880119899) int119880119899 119891119889 (]10158401015840)

=WMV119880120583 (119891)

(46)



(1) Θ(120583 ]) isin [0 1](2) Θ(120583 ]) = 1 minus Θ(] 120583)









120588 (119880119899)120583 (119880119899) =120588 (119880119899)] (119880119899)

] (119880119899)120583 (119880119899) (48)


120588 (119880119899)120583 (119880119899) =(120583 + 120588) (119880119899)120583 (119880119899) minus 1

120588 (119880119899)] (119880119899) = (] + 120588) (119880119899) ] (119880119899) minus 1] (119880119899)120583 (119880119899) = (120583 + ]) (119880119899) 120583 (119880119899) minus 1

(49)

and we get

120583 (119880119899)(120583 + 120588) (119880119899) =(120583 (119880119899) (120583 + ]) (119880119899)) (] (119880119899) (] + 120588) (119880119899))2 (120583 (119880119899) (120583 + ]) (119880119899)) (] (119880119899) (] + 120588) (119880119899)) minus 120583 (119880119899) (120583 + ]) (119880119899) minus ] (119880119899) (] + 120588) (119880119899) + 1 (50)










lim119909rarrinfin

119891 (119909) = 0 (51)


thussup119909isin119870119888119899



120583 (1198701198990) lt 12059810158401015840120583 (119870119899)



100381610038161003816100381610038161003816100381610038161003816 le int119870119899 10038161003816100381610038161198911003816100381610038161003816 119889120583 = int119870119899010038161003816100381610038161198911003816100381610038161003816 119889120583 + int

119870119899minus1198701198990

10038161003816100381610038161198911003816100381610038161003816 119889120583le (sup

119883

10038161003816100381610038161198911003816100381610038161003816) 120583 (1198701198990) + 1205981015840120583 (119870119899 minus 1198701198990

) (54)



and hence WMV119870120583 (119891) = 0


120582 (1R) = 1 andWMV119870





119901 (119882119872119881119880120583 (1198911)) minus 120598 le 119901 (119882119872119881119880

120583 (1198912))le 119901 (119882119872119881119880

120583 (1198911)) + 120598 (56)



le 120598 + 119901( 1120583 (119880119899) int119880119899 1198911119889120583)

(57)


119901( 1120583 (119880119899) int119880119899 1198911119889120583) minus 120598 le 119901(1120583 (119880119899) int119880119899 1198912119889120583) (58)




(1) 119891 isin F119880120583

(2) 119882119872119881119880120583 (119891) = lim119899rarr+infin119882119872119881119880

120583 (119891119899)Proof Let 119906119899 =WMV119880



each (119899119898) isin N2

sup119909isin119883

119901 (119891119899 minus 119891119898) lt 120598 (59)


120583 (119891119899 minus 119891119898)) le 120598 (60)



(sup119909isin119883


119901 (119891119898 minus 119891) lt 120598)997904rArr sup

119909isin119883119901 (119891119899 minus 119891119898) lt 2120598

997904rArr 119901(WMV119880120583 (119891119899 minus 119891119898)) lt 2120598

997904rArr 119901 (119906119899 minus 119906) lt 2120598

(61)





(62)


Then

119901( 1120583 (119880119899) int119880119899 119891119889120583 minus1120583 (119880119899) int119880119899 119891119896119889120583) lt

1205988 119901 (119906119896 minus 119906) lt 1205984

(63)


119901( 1120583 (119880119899) int119880119899 119891119870+1119889120583 minus 119906119870+1) lt1205988 (64)


31205988 (65)


119901( 1120583 (119880119899) int119880119899 119891119889120583 minus 119906) lt1205988 + 31205988 + 1205984 lt 120598 (66)


119870119899 = [minus119899 minus 1 119899 + 1]119899 (67)


119871119899 = 119909 isin R119899 119909 le 119899 + 1 (68)




lim119899rarr+infin

120582 (119880119899Δ119880119899 + V)120582 (119880119899) = 0 (69)

then


120582 ) and 119882119872119881119880120582 (119891) =119882119872119881119880+V


120582 and119882119872119881119880120582 (119891) = 119882119872119881119880


120582 (119891) =WMV119870120582 (119891V)


1120582 (119880119899) int119880119899 119891119889120582 minus1120582 (119880119899 + V) int119880119899+V 119891119889120582 =

1120582 (119880119899)sdot int

119880119899


(70)


1120582 (119880119899 + V) int119880119899+V 119891119889120582100381610038161003816100381610038161003816100381610038161003816

le 119872120582 (119880119899Δ119880119899 + V)120582 (119880119899) (71)



lim119899rarr+infin

120582 (119870119899Δ119870119899 + V)120582 (119870119899) = 0lim

119899rarr+infin

120582 (119871119899Δ119871119899 + V)120582 (119871119899) = 0(72)


120582 (119870119899Δ119870119899 + V) = (119899 + 1)119898 120582 (1198700Δ1198700 + 1119899 + 1V) (73)

Let



lim119899rarr+infin

120582 (119870119899Δ119870119899 + V)120582 (119870119899) = 0 (75)





120582 ) and119882119872119881119870120582 (119891) =119882119872119881119870+V

120582 (119891) (resp119882119872119881119871120582 (119891) = 119882119872119881119871+V


120582 (resp 119891 isin F119871120582)

and 119882119872119881119870120582 (119891) = 119882119872119881119870

120582 (119891V) (resp 119882119872119881119871120582 (119891) =119882119872119881119871





119868119891120583 (119891 119905) = 1120583 (119865 le 119905) int119865le119905 119891 (119909) 119889120583 (119909) (77)


WMV119865120583 (119891) = lim

119905rarr+infin119868119865120583 (119891 119905) (78)


WMV119880120583 (119891) =WMV119865

120583 (119891) (79)



119865 = 119905 = 119865minus1 (119905) (80)





where

119868 (120572) 119904 simplex 997891997888rarr 119868 (120572) (119904) = int119904120572 (82)



120593119904 = |119868 (sdot) (119904)|1 + |119868 (sdot) (119904)| (83)


120582 (120593119904) = 0 (84)


119904120572 = 0 thus 120593119904(120572) = 0 Finally

WMV119880120582 (120593119904) = 0 (85)


119904120572119904 = 1 Let 120587119904 be



10038161003816100381610038161003816100381610038161003816 gt 12 (86)

Then


Moreover

inf120572isin119881119899

120593119904 (120572) = 12 120582 (119881119899) gt 120582 (119880119899minus1)

(88)

Then

int119880119899

120593119904119889120582 ge int119881119899

120593119904119889120582 ge 120582 (119881119899)2 (89)


Thus

WMV119880120582 (120593119904) = lim

119899rarr+infin

1120582 (119880119899) int119880119899 120593119904119889120582ge lim

119899rarr+infin

120582 (119881119899)2120582 (119880119899) ge lim119899rarr+infin

120582 (119880119899minus1)2120582 (119880119899) =12

= 0(90)


120582 (120593119904) is a 0 1-valued map




119891 ((119909120582)Λ) = 119891 ((119909120582)Λ) forall (119909120582)Λ isin 119883 (91)



(119891) (92)





11988012058211205831205821


we have


(119891) = prod120582isinΛ

119882119872119881119880120582120583120582(119891120582) (93)





(⨂120582isinΛ

120583120582)((prod120582isinΛ


(94)

where


(95)








each (119899119898) isin N2

sup119909isin119883

119901 (119891119899 minus 119891119898) lt 120598 (96)


120583 (119891119899 minus 119891119898)) le 120598 (97)




120583 (1198911015840119899)



11989110158401015840119899 =

11989110158401198992 if 119899 is even

119891(119899minus1)2 if 119899 is odd (98)



2119899+1)119899isinN and(1198911015840119899)119899isinN = (11989110158401015840

2119899+1)119899isinN we get

1199061015840 = lim119899rarr+infin

WMV119880120583 (11989110158401015840

119899 )= lim


120583 (119891119899) = 119906lim


120583 (1198911015840119899) = lim


120583 (11989110158401015840119899 ) = 119906

(99)



WMV119880120583 (119891) = lim


120583 (119891119899) (100)


of Proposition 18



2 119899= 119898 997904rArr 119906119899 = 119906119898 (101)




2 119899 = 119898 997904rArr 119906119899 = 119906119898 (102)



2 (1 le 119899 lt 119898 le 119896) 997904rArr (119906119899 = 119906119898) (103)


120583otimes119899 (119880119899119896 ) = 120583otimes119899 ((1199091 119909119899)



WMV119880120583 (119891) =WMV119880Ord(119891)

120583otimesOrd(119891) (119891) (105)









119891 (119909) = 119891 ∘ 120587 (119909) forall119909 isin 119865 (107)




119891119899 ∘ 120587119898 = 119891119899 forall119899 le 119898 (108)




119865119891119899+1 = 119865119891119899 + 119865 (109)


This ends the proof


(prod120582isinΛ

119880120582119899) minusDΛ119899119909 = 0 (111)


WMV120582 (119891) (112)





119882119872119881120582 (119891120572) = 119882119872119881120582 (119891) (113)


WMV120582 ((119891119899)120572) =WMV120572minus119898120582 (119891119899) =WMV120582 (119891119899) (114)


119882119872119881120582 (119891120572) = 119882119872119881120582 (119891) (115)


119882119872119881120582 (119891V) = 119882119872119881120582 (119891) (116)



WMV120582 ((119891119899)V119899)= lim


=WMV120582 (119891) (117)


119882119872119881120582 (119891 ∘ 119906) = 119882119872119881120582 (119891) (118)






⨁119899isinN

119865119891119899 sub 119866 (120)


lim119899rarr+infin

119892119899 = 119892 (121)



WMV120582 (119892) =WMV120582 (119891) (123)


Competing Interests


References





























Volume 2014




Journal of











Journal of


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Algebra










and that


119905] (119880119899)(119905] + (1 minus 119905) ]1015840) (119880119899)sdot 1

] (119880119899) int119880119899 119891119889 (]) minusWMV119880120583 (119891)

(45)




120583 (119891) = 0 the same way Thus

WMV119880]10158401015840 (119891) = lim

119899rarr+infin

1]10158401015840 (119880119899) int119880119899 119891119889 (]10158401015840)

=WMV119880120583 (119891)

(46)



(1) Θ(120583 ]) isin [0 1](2) Θ(120583 ]) = 1 minus Θ(] 120583)









120588 (119880119899)120583 (119880119899) =120588 (119880119899)] (119880119899)

] (119880119899)120583 (119880119899) (48)


120588 (119880119899)120583 (119880119899) =(120583 + 120588) (119880119899)120583 (119880119899) minus 1

120588 (119880119899)] (119880119899) = (] + 120588) (119880119899) ] (119880119899) minus 1] (119880119899)120583 (119880119899) = (120583 + ]) (119880119899) 120583 (119880119899) minus 1

(49)

and we get

120583 (119880119899)(120583 + 120588) (119880119899) =(120583 (119880119899) (120583 + ]) (119880119899)) (] (119880119899) (] + 120588) (119880119899))2 (120583 (119880119899) (120583 + ]) (119880119899)) (] (119880119899) (] + 120588) (119880119899)) minus 120583 (119880119899) (120583 + ]) (119880119899) minus ] (119880119899) (] + 120588) (119880119899) + 1 (50)










lim119909rarrinfin

119891 (119909) = 0 (51)


thussup119909isin119870119888119899



120583 (1198701198990) lt 12059810158401015840120583 (119870119899)



100381610038161003816100381610038161003816100381610038161003816 le int119870119899 10038161003816100381610038161198911003816100381610038161003816 119889120583 = int119870119899010038161003816100381610038161198911003816100381610038161003816 119889120583 + int

119870119899minus1198701198990

10038161003816100381610038161198911003816100381610038161003816 119889120583le (sup

119883

10038161003816100381610038161198911003816100381610038161003816) 120583 (1198701198990) + 1205981015840120583 (119870119899 minus 1198701198990

) (54)



and hence WMV119870120583 (119891) = 0


120582 (1R) = 1 andWMV119870





119901 (119882119872119881119880120583 (1198911)) minus 120598 le 119901 (119882119872119881119880

120583 (1198912))le 119901 (119882119872119881119880

120583 (1198911)) + 120598 (56)



le 120598 + 119901( 1120583 (119880119899) int119880119899 1198911119889120583)

(57)


119901( 1120583 (119880119899) int119880119899 1198911119889120583) minus 120598 le 119901(1120583 (119880119899) int119880119899 1198912119889120583) (58)




(1) 119891 isin F119880120583

(2) 119882119872119881119880120583 (119891) = lim119899rarr+infin119882119872119881119880

120583 (119891119899)Proof Let 119906119899 =WMV119880



each (119899119898) isin N2

sup119909isin119883

119901 (119891119899 minus 119891119898) lt 120598 (59)


120583 (119891119899 minus 119891119898)) le 120598 (60)



(sup119909isin119883


119901 (119891119898 minus 119891) lt 120598)997904rArr sup

119909isin119883119901 (119891119899 minus 119891119898) lt 2120598

997904rArr 119901(WMV119880120583 (119891119899 minus 119891119898)) lt 2120598

997904rArr 119901 (119906119899 minus 119906) lt 2120598

(61)





(62)


Then

119901( 1120583 (119880119899) int119880119899 119891119889120583 minus1120583 (119880119899) int119880119899 119891119896119889120583) lt

1205988 119901 (119906119896 minus 119906) lt 1205984

(63)


119901( 1120583 (119880119899) int119880119899 119891119870+1119889120583 minus 119906119870+1) lt1205988 (64)


31205988 (65)


119901( 1120583 (119880119899) int119880119899 119891119889120583 minus 119906) lt1205988 + 31205988 + 1205984 lt 120598 (66)


119870119899 = [minus119899 minus 1 119899 + 1]119899 (67)


119871119899 = 119909 isin R119899 119909 le 119899 + 1 (68)




lim119899rarr+infin

120582 (119880119899Δ119880119899 + V)120582 (119880119899) = 0 (69)

then


120582 ) and 119882119872119881119880120582 (119891) =119882119872119881119880+V


120582 and119882119872119881119880120582 (119891) = 119882119872119881119880


120582 (119891) =WMV119870120582 (119891V)


1120582 (119880119899) int119880119899 119891119889120582 minus1120582 (119880119899 + V) int119880119899+V 119891119889120582 =

1120582 (119880119899)sdot int

119880119899


(70)


1120582 (119880119899 + V) int119880119899+V 119891119889120582100381610038161003816100381610038161003816100381610038161003816

le 119872120582 (119880119899Δ119880119899 + V)120582 (119880119899) (71)



lim119899rarr+infin

120582 (119870119899Δ119870119899 + V)120582 (119870119899) = 0lim

119899rarr+infin

120582 (119871119899Δ119871119899 + V)120582 (119871119899) = 0(72)


120582 (119870119899Δ119870119899 + V) = (119899 + 1)119898 120582 (1198700Δ1198700 + 1119899 + 1V) (73)

Let



lim119899rarr+infin

120582 (119870119899Δ119870119899 + V)120582 (119870119899) = 0 (75)





120582 ) and119882119872119881119870120582 (119891) =119882119872119881119870+V

120582 (119891) (resp119882119872119881119871120582 (119891) = 119882119872119881119871+V


120582 (resp 119891 isin F119871120582)

and 119882119872119881119870120582 (119891) = 119882119872119881119870

120582 (119891V) (resp 119882119872119881119871120582 (119891) =119882119872119881119871





119868119891120583 (119891 119905) = 1120583 (119865 le 119905) int119865le119905 119891 (119909) 119889120583 (119909) (77)


WMV119865120583 (119891) = lim

119905rarr+infin119868119865120583 (119891 119905) (78)


WMV119880120583 (119891) =WMV119865

120583 (119891) (79)



119865 = 119905 = 119865minus1 (119905) (80)





where

119868 (120572) 119904 simplex 997891997888rarr 119868 (120572) (119904) = int119904120572 (82)



120593119904 = |119868 (sdot) (119904)|1 + |119868 (sdot) (119904)| (83)


120582 (120593119904) = 0 (84)


119904120572 = 0 thus 120593119904(120572) = 0 Finally

WMV119880120582 (120593119904) = 0 (85)


119904120572119904 = 1 Let 120587119904 be



10038161003816100381610038161003816100381610038161003816 gt 12 (86)

Then


Moreover

inf120572isin119881119899

120593119904 (120572) = 12 120582 (119881119899) gt 120582 (119880119899minus1)

(88)

Then

int119880119899

120593119904119889120582 ge int119881119899

120593119904119889120582 ge 120582 (119881119899)2 (89)


Thus

WMV119880120582 (120593119904) = lim

119899rarr+infin

1120582 (119880119899) int119880119899 120593119904119889120582ge lim

119899rarr+infin

120582 (119881119899)2120582 (119880119899) ge lim119899rarr+infin

120582 (119880119899minus1)2120582 (119880119899) =12

= 0(90)


120582 (120593119904) is a 0 1-valued map




119891 ((119909120582)Λ) = 119891 ((119909120582)Λ) forall (119909120582)Λ isin 119883 (91)



(119891) (92)





11988012058211205831205821


we have


(119891) = prod120582isinΛ

119882119872119881119880120582120583120582(119891120582) (93)





(⨂120582isinΛ

120583120582)((prod120582isinΛ


(94)

where


(95)








each (119899119898) isin N2

sup119909isin119883

119901 (119891119899 minus 119891119898) lt 120598 (96)


120583 (119891119899 minus 119891119898)) le 120598 (97)




120583 (1198911015840119899)



11989110158401015840119899 =

11989110158401198992 if 119899 is even

119891(119899minus1)2 if 119899 is odd (98)



2119899+1)119899isinN and(1198911015840119899)119899isinN = (11989110158401015840

2119899+1)119899isinN we get

1199061015840 = lim119899rarr+infin

WMV119880120583 (11989110158401015840

119899 )= lim


120583 (119891119899) = 119906lim


120583 (1198911015840119899) = lim


120583 (11989110158401015840119899 ) = 119906

(99)



WMV119880120583 (119891) = lim


120583 (119891119899) (100)


of Proposition 18



2 119899= 119898 997904rArr 119906119899 = 119906119898 (101)




2 119899 = 119898 997904rArr 119906119899 = 119906119898 (102)



2 (1 le 119899 lt 119898 le 119896) 997904rArr (119906119899 = 119906119898) (103)


120583otimes119899 (119880119899119896 ) = 120583otimes119899 ((1199091 119909119899)



WMV119880120583 (119891) =WMV119880Ord(119891)

120583otimesOrd(119891) (119891) (105)









119891 (119909) = 119891 ∘ 120587 (119909) forall119909 isin 119865 (107)




119891119899 ∘ 120587119898 = 119891119899 forall119899 le 119898 (108)




119865119891119899+1 = 119865119891119899 + 119865 (109)


This ends the proof


(prod120582isinΛ

119880120582119899) minusDΛ119899119909 = 0 (111)


WMV120582 (119891) (112)





119882119872119881120582 (119891120572) = 119882119872119881120582 (119891) (113)


WMV120582 ((119891119899)120572) =WMV120572minus119898120582 (119891119899) =WMV120582 (119891119899) (114)


119882119872119881120582 (119891120572) = 119882119872119881120582 (119891) (115)


119882119872119881120582 (119891V) = 119882119872119881120582 (119891) (116)



WMV120582 ((119891119899)V119899)= lim


=WMV120582 (119891) (117)


119882119872119881120582 (119891 ∘ 119906) = 119882119872119881120582 (119891) (118)






⨁119899isinN

119865119891119899 sub 119866 (120)


lim119899rarr+infin

119892119899 = 119892 (121)



WMV120582 (119892) =WMV120582 (119891) (123)


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Journal of


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lim119909rarrinfin

119891 (119909) = 0 (51)


thussup119909isin119870119888119899



120583 (1198701198990) lt 12059810158401015840120583 (119870119899)



100381610038161003816100381610038161003816100381610038161003816 le int119870119899 10038161003816100381610038161198911003816100381610038161003816 119889120583 = int119870119899010038161003816100381610038161198911003816100381610038161003816 119889120583 + int

119870119899minus1198701198990

10038161003816100381610038161198911003816100381610038161003816 119889120583le (sup

119883

10038161003816100381610038161198911003816100381610038161003816) 120583 (1198701198990) + 1205981015840120583 (119870119899 minus 1198701198990

) (54)



and hence WMV119870120583 (119891) = 0


120582 (1R) = 1 andWMV119870





119901 (119882119872119881119880120583 (1198911)) minus 120598 le 119901 (119882119872119881119880

120583 (1198912))le 119901 (119882119872119881119880

120583 (1198911)) + 120598 (56)



le 120598 + 119901( 1120583 (119880119899) int119880119899 1198911119889120583)

(57)


119901( 1120583 (119880119899) int119880119899 1198911119889120583) minus 120598 le 119901(1120583 (119880119899) int119880119899 1198912119889120583) (58)




(1) 119891 isin F119880120583

(2) 119882119872119881119880120583 (119891) = lim119899rarr+infin119882119872119881119880

120583 (119891119899)Proof Let 119906119899 =WMV119880



each (119899119898) isin N2

sup119909isin119883

119901 (119891119899 minus 119891119898) lt 120598 (59)


120583 (119891119899 minus 119891119898)) le 120598 (60)



(sup119909isin119883


119901 (119891119898 minus 119891) lt 120598)997904rArr sup

119909isin119883119901 (119891119899 minus 119891119898) lt 2120598

997904rArr 119901(WMV119880120583 (119891119899 minus 119891119898)) lt 2120598

997904rArr 119901 (119906119899 minus 119906) lt 2120598

(61)





(62)


Then

119901( 1120583 (119880119899) int119880119899 119891119889120583 minus1120583 (119880119899) int119880119899 119891119896119889120583) lt

1205988 119901 (119906119896 minus 119906) lt 1205984

(63)


119901( 1120583 (119880119899) int119880119899 119891119870+1119889120583 minus 119906119870+1) lt1205988 (64)


31205988 (65)


119901( 1120583 (119880119899) int119880119899 119891119889120583 minus 119906) lt1205988 + 31205988 + 1205984 lt 120598 (66)


119870119899 = [minus119899 minus 1 119899 + 1]119899 (67)


119871119899 = 119909 isin R119899 119909 le 119899 + 1 (68)




lim119899rarr+infin

120582 (119880119899Δ119880119899 + V)120582 (119880119899) = 0 (69)

then


120582 ) and 119882119872119881119880120582 (119891) =119882119872119881119880+V


120582 and119882119872119881119880120582 (119891) = 119882119872119881119880


120582 (119891) =WMV119870120582 (119891V)


1120582 (119880119899) int119880119899 119891119889120582 minus1120582 (119880119899 + V) int119880119899+V 119891119889120582 =

1120582 (119880119899)sdot int

119880119899


(70)


1120582 (119880119899 + V) int119880119899+V 119891119889120582100381610038161003816100381610038161003816100381610038161003816

le 119872120582 (119880119899Δ119880119899 + V)120582 (119880119899) (71)



lim119899rarr+infin

120582 (119870119899Δ119870119899 + V)120582 (119870119899) = 0lim

119899rarr+infin

120582 (119871119899Δ119871119899 + V)120582 (119871119899) = 0(72)


120582 (119870119899Δ119870119899 + V) = (119899 + 1)119898 120582 (1198700Δ1198700 + 1119899 + 1V) (73)

Let



lim119899rarr+infin

120582 (119870119899Δ119870119899 + V)120582 (119870119899) = 0 (75)





120582 ) and119882119872119881119870120582 (119891) =119882119872119881119870+V

120582 (119891) (resp119882119872119881119871120582 (119891) = 119882119872119881119871+V


120582 (resp 119891 isin F119871120582)

and 119882119872119881119870120582 (119891) = 119882119872119881119870

120582 (119891V) (resp 119882119872119881119871120582 (119891) =119882119872119881119871





119868119891120583 (119891 119905) = 1120583 (119865 le 119905) int119865le119905 119891 (119909) 119889120583 (119909) (77)


WMV119865120583 (119891) = lim

119905rarr+infin119868119865120583 (119891 119905) (78)


WMV119880120583 (119891) =WMV119865

120583 (119891) (79)



119865 = 119905 = 119865minus1 (119905) (80)





where

119868 (120572) 119904 simplex 997891997888rarr 119868 (120572) (119904) = int119904120572 (82)



120593119904 = |119868 (sdot) (119904)|1 + |119868 (sdot) (119904)| (83)


120582 (120593119904) = 0 (84)


119904120572 = 0 thus 120593119904(120572) = 0 Finally

WMV119880120582 (120593119904) = 0 (85)


119904120572119904 = 1 Let 120587119904 be



10038161003816100381610038161003816100381610038161003816 gt 12 (86)

Then


Moreover

inf120572isin119881119899

120593119904 (120572) = 12 120582 (119881119899) gt 120582 (119880119899minus1)

(88)

Then

int119880119899

120593119904119889120582 ge int119881119899

120593119904119889120582 ge 120582 (119881119899)2 (89)


Thus

WMV119880120582 (120593119904) = lim

119899rarr+infin

1120582 (119880119899) int119880119899 120593119904119889120582ge lim

119899rarr+infin

120582 (119881119899)2120582 (119880119899) ge lim119899rarr+infin

120582 (119880119899minus1)2120582 (119880119899) =12

= 0(90)


120582 (120593119904) is a 0 1-valued map




119891 ((119909120582)Λ) = 119891 ((119909120582)Λ) forall (119909120582)Λ isin 119883 (91)



(119891) (92)





11988012058211205831205821


we have


(119891) = prod120582isinΛ

119882119872119881119880120582120583120582(119891120582) (93)





(⨂120582isinΛ

120583120582)((prod120582isinΛ


(94)

where


(95)








each (119899119898) isin N2

sup119909isin119883

119901 (119891119899 minus 119891119898) lt 120598 (96)


120583 (119891119899 minus 119891119898)) le 120598 (97)




120583 (1198911015840119899)



11989110158401015840119899 =

11989110158401198992 if 119899 is even

119891(119899minus1)2 if 119899 is odd (98)



2119899+1)119899isinN and(1198911015840119899)119899isinN = (11989110158401015840

2119899+1)119899isinN we get

1199061015840 = lim119899rarr+infin

WMV119880120583 (11989110158401015840

119899 )= lim


120583 (119891119899) = 119906lim


120583 (1198911015840119899) = lim


120583 (11989110158401015840119899 ) = 119906

(99)



WMV119880120583 (119891) = lim


120583 (119891119899) (100)


of Proposition 18



2 119899= 119898 997904rArr 119906119899 = 119906119898 (101)




2 119899 = 119898 997904rArr 119906119899 = 119906119898 (102)



2 (1 le 119899 lt 119898 le 119896) 997904rArr (119906119899 = 119906119898) (103)


120583otimes119899 (119880119899119896 ) = 120583otimes119899 ((1199091 119909119899)



WMV119880120583 (119891) =WMV119880Ord(119891)

120583otimesOrd(119891) (119891) (105)









119891 (119909) = 119891 ∘ 120587 (119909) forall119909 isin 119865 (107)




119891119899 ∘ 120587119898 = 119891119899 forall119899 le 119898 (108)




119865119891119899+1 = 119865119891119899 + 119865 (109)


This ends the proof


(prod120582isinΛ

119880120582119899) minusDΛ119899119909 = 0 (111)


WMV120582 (119891) (112)





119882119872119881120582 (119891120572) = 119882119872119881120582 (119891) (113)


WMV120582 ((119891119899)120572) =WMV120572minus119898120582 (119891119899) =WMV120582 (119891119899) (114)


119882119872119881120582 (119891120572) = 119882119872119881120582 (119891) (115)


119882119872119881120582 (119891V) = 119882119872119881120582 (119891) (116)



WMV120582 ((119891119899)V119899)= lim


=WMV120582 (119891) (117)


119882119872119881120582 (119891 ∘ 119906) = 119882119872119881120582 (119891) (118)






⨁119899isinN

119865119891119899 sub 119866 (120)


lim119899rarr+infin

119892119899 = 119892 (121)



WMV120582 (119892) =WMV120582 (119891) (123)


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Volume 2014




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Journal of


Function Spaces






Algebra













(62)


Then

119901( 1120583 (119880119899) int119880119899 119891119889120583 minus1120583 (119880119899) int119880119899 119891119896119889120583) lt

1205988 119901 (119906119896 minus 119906) lt 1205984

(63)


119901( 1120583 (119880119899) int119880119899 119891119870+1119889120583 minus 119906119870+1) lt1205988 (64)


31205988 (65)


119901( 1120583 (119880119899) int119880119899 119891119889120583 minus 119906) lt1205988 + 31205988 + 1205984 lt 120598 (66)


119870119899 = [minus119899 minus 1 119899 + 1]119899 (67)


119871119899 = 119909 isin R119899 119909 le 119899 + 1 (68)




lim119899rarr+infin

120582 (119880119899Δ119880119899 + V)120582 (119880119899) = 0 (69)

then


120582 ) and 119882119872119881119880120582 (119891) =119882119872119881119880+V


120582 and119882119872119881119880120582 (119891) = 119882119872119881119880


120582 (119891) =WMV119870120582 (119891V)


1120582 (119880119899) int119880119899 119891119889120582 minus1120582 (119880119899 + V) int119880119899+V 119891119889120582 =

1120582 (119880119899)sdot int

119880119899


(70)


1120582 (119880119899 + V) int119880119899+V 119891119889120582100381610038161003816100381610038161003816100381610038161003816

le 119872120582 (119880119899Δ119880119899 + V)120582 (119880119899) (71)



lim119899rarr+infin

120582 (119870119899Δ119870119899 + V)120582 (119870119899) = 0lim

119899rarr+infin

120582 (119871119899Δ119871119899 + V)120582 (119871119899) = 0(72)


120582 (119870119899Δ119870119899 + V) = (119899 + 1)119898 120582 (1198700Δ1198700 + 1119899 + 1V) (73)

Let



lim119899rarr+infin

120582 (119870119899Δ119870119899 + V)120582 (119870119899) = 0 (75)





120582 ) and119882119872119881119870120582 (119891) =119882119872119881119870+V

120582 (119891) (resp119882119872119881119871120582 (119891) = 119882119872119881119871+V


120582 (resp 119891 isin F119871120582)

and 119882119872119881119870120582 (119891) = 119882119872119881119870

120582 (119891V) (resp 119882119872119881119871120582 (119891) =119882119872119881119871





119868119891120583 (119891 119905) = 1120583 (119865 le 119905) int119865le119905 119891 (119909) 119889120583 (119909) (77)


WMV119865120583 (119891) = lim

119905rarr+infin119868119865120583 (119891 119905) (78)


WMV119880120583 (119891) =WMV119865

120583 (119891) (79)



119865 = 119905 = 119865minus1 (119905) (80)





where

119868 (120572) 119904 simplex 997891997888rarr 119868 (120572) (119904) = int119904120572 (82)



120593119904 = |119868 (sdot) (119904)|1 + |119868 (sdot) (119904)| (83)


120582 (120593119904) = 0 (84)


119904120572 = 0 thus 120593119904(120572) = 0 Finally

WMV119880120582 (120593119904) = 0 (85)


119904120572119904 = 1 Let 120587119904 be



10038161003816100381610038161003816100381610038161003816 gt 12 (86)

Then


Moreover

inf120572isin119881119899

120593119904 (120572) = 12 120582 (119881119899) gt 120582 (119880119899minus1)

(88)

Then

int119880119899

120593119904119889120582 ge int119881119899

120593119904119889120582 ge 120582 (119881119899)2 (89)


Thus

WMV119880120582 (120593119904) = lim

119899rarr+infin

1120582 (119880119899) int119880119899 120593119904119889120582ge lim

119899rarr+infin

120582 (119881119899)2120582 (119880119899) ge lim119899rarr+infin

120582 (119880119899minus1)2120582 (119880119899) =12

= 0(90)


120582 (120593119904) is a 0 1-valued map




119891 ((119909120582)Λ) = 119891 ((119909120582)Λ) forall (119909120582)Λ isin 119883 (91)



(119891) (92)





11988012058211205831205821


we have


(119891) = prod120582isinΛ

119882119872119881119880120582120583120582(119891120582) (93)





(⨂120582isinΛ

120583120582)((prod120582isinΛ


(94)

where


(95)








each (119899119898) isin N2

sup119909isin119883

119901 (119891119899 minus 119891119898) lt 120598 (96)


120583 (119891119899 minus 119891119898)) le 120598 (97)




120583 (1198911015840119899)



11989110158401015840119899 =

11989110158401198992 if 119899 is even

119891(119899minus1)2 if 119899 is odd (98)



2119899+1)119899isinN and(1198911015840119899)119899isinN = (11989110158401015840

2119899+1)119899isinN we get

1199061015840 = lim119899rarr+infin

WMV119880120583 (11989110158401015840

119899 )= lim


120583 (119891119899) = 119906lim


120583 (1198911015840119899) = lim


120583 (11989110158401015840119899 ) = 119906

(99)



WMV119880120583 (119891) = lim


120583 (119891119899) (100)


of Proposition 18



2 119899= 119898 997904rArr 119906119899 = 119906119898 (101)




2 119899 = 119898 997904rArr 119906119899 = 119906119898 (102)



2 (1 le 119899 lt 119898 le 119896) 997904rArr (119906119899 = 119906119898) (103)


120583otimes119899 (119880119899119896 ) = 120583otimes119899 ((1199091 119909119899)



WMV119880120583 (119891) =WMV119880Ord(119891)

120583otimesOrd(119891) (119891) (105)









119891 (119909) = 119891 ∘ 120587 (119909) forall119909 isin 119865 (107)




119891119899 ∘ 120587119898 = 119891119899 forall119899 le 119898 (108)




119865119891119899+1 = 119865119891119899 + 119865 (109)


This ends the proof


(prod120582isinΛ

119880120582119899) minusDΛ119899119909 = 0 (111)


WMV120582 (119891) (112)





119882119872119881120582 (119891120572) = 119882119872119881120582 (119891) (113)


WMV120582 ((119891119899)120572) =WMV120572minus119898120582 (119891119899) =WMV120582 (119891119899) (114)


119882119872119881120582 (119891120572) = 119882119872119881120582 (119891) (115)


119882119872119881120582 (119891V) = 119882119872119881120582 (119891) (116)



WMV120582 ((119891119899)V119899)= lim


=WMV120582 (119891) (117)


119882119872119881120582 (119891 ∘ 119906) = 119882119872119881120582 (119891) (118)






⨁119899isinN

119865119891119899 sub 119866 (120)


lim119899rarr+infin

119892119899 = 119892 (121)



WMV120582 (119892) =WMV120582 (119891) (123)


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References





























Volume 2014




Journal of











Journal of


Function Spaces






Algebra













120582 ) and119882119872119881119870120582 (119891) =119882119872119881119870+V

120582 (119891) (resp119882119872119881119871120582 (119891) = 119882119872119881119871+V


120582 (resp 119891 isin F119871120582)

and 119882119872119881119870120582 (119891) = 119882119872119881119870

120582 (119891V) (resp 119882119872119881119871120582 (119891) =119882119872119881119871





119868119891120583 (119891 119905) = 1120583 (119865 le 119905) int119865le119905 119891 (119909) 119889120583 (119909) (77)


WMV119865120583 (119891) = lim

119905rarr+infin119868119865120583 (119891 119905) (78)


WMV119880120583 (119891) =WMV119865

120583 (119891) (79)



119865 = 119905 = 119865minus1 (119905) (80)





where

119868 (120572) 119904 simplex 997891997888rarr 119868 (120572) (119904) = int119904120572 (82)



120593119904 = |119868 (sdot) (119904)|1 + |119868 (sdot) (119904)| (83)


120582 (120593119904) = 0 (84)


119904120572 = 0 thus 120593119904(120572) = 0 Finally

WMV119880120582 (120593119904) = 0 (85)


119904120572119904 = 1 Let 120587119904 be



10038161003816100381610038161003816100381610038161003816 gt 12 (86)

Then


Moreover

inf120572isin119881119899

120593119904 (120572) = 12 120582 (119881119899) gt 120582 (119880119899minus1)

(88)

Then

int119880119899

120593119904119889120582 ge int119881119899

120593119904119889120582 ge 120582 (119881119899)2 (89)


Thus

WMV119880120582 (120593119904) = lim

119899rarr+infin

1120582 (119880119899) int119880119899 120593119904119889120582ge lim

119899rarr+infin

120582 (119881119899)2120582 (119880119899) ge lim119899rarr+infin

120582 (119880119899minus1)2120582 (119880119899) =12

= 0(90)


120582 (120593119904) is a 0 1-valued map




119891 ((119909120582)Λ) = 119891 ((119909120582)Λ) forall (119909120582)Λ isin 119883 (91)



(119891) (92)





11988012058211205831205821


we have


(119891) = prod120582isinΛ

119882119872119881119880120582120583120582(119891120582) (93)





(⨂120582isinΛ

120583120582)((prod120582isinΛ


(94)

where


(95)








each (119899119898) isin N2

sup119909isin119883

119901 (119891119899 minus 119891119898) lt 120598 (96)


120583 (119891119899 minus 119891119898)) le 120598 (97)




120583 (1198911015840119899)



11989110158401015840119899 =

11989110158401198992 if 119899 is even

119891(119899minus1)2 if 119899 is odd (98)



2119899+1)119899isinN and(1198911015840119899)119899isinN = (11989110158401015840

2119899+1)119899isinN we get

1199061015840 = lim119899rarr+infin

WMV119880120583 (11989110158401015840

119899 )= lim


120583 (119891119899) = 119906lim


120583 (1198911015840119899) = lim


120583 (11989110158401015840119899 ) = 119906

(99)



WMV119880120583 (119891) = lim


120583 (119891119899) (100)


of Proposition 18



2 119899= 119898 997904rArr 119906119899 = 119906119898 (101)




2 119899 = 119898 997904rArr 119906119899 = 119906119898 (102)



2 (1 le 119899 lt 119898 le 119896) 997904rArr (119906119899 = 119906119898) (103)


120583otimes119899 (119880119899119896 ) = 120583otimes119899 ((1199091 119909119899)



WMV119880120583 (119891) =WMV119880Ord(119891)

120583otimesOrd(119891) (119891) (105)









119891 (119909) = 119891 ∘ 120587 (119909) forall119909 isin 119865 (107)




119891119899 ∘ 120587119898 = 119891119899 forall119899 le 119898 (108)




119865119891119899+1 = 119865119891119899 + 119865 (109)


This ends the proof


(prod120582isinΛ

119880120582119899) minusDΛ119899119909 = 0 (111)


WMV120582 (119891) (112)





119882119872119881120582 (119891120572) = 119882119872119881120582 (119891) (113)


WMV120582 ((119891119899)120572) =WMV120572minus119898120582 (119891119899) =WMV120582 (119891119899) (114)


119882119872119881120582 (119891120572) = 119882119872119881120582 (119891) (115)


119882119872119881120582 (119891V) = 119882119872119881120582 (119891) (116)



WMV120582 ((119891119899)V119899)= lim


=WMV120582 (119891) (117)


119882119872119881120582 (119891 ∘ 119906) = 119882119872119881120582 (119891) (118)






⨁119899isinN

119865119891119899 sub 119866 (120)


lim119899rarr+infin

119892119899 = 119892 (121)



WMV120582 (119892) =WMV120582 (119891) (123)


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Thus

WMV119880120582 (120593119904) = lim

119899rarr+infin

1120582 (119880119899) int119880119899 120593119904119889120582ge lim

119899rarr+infin

120582 (119881119899)2120582 (119880119899) ge lim119899rarr+infin

120582 (119880119899minus1)2120582 (119880119899) =12

= 0(90)


120582 (120593119904) is a 0 1-valued map




119891 ((119909120582)Λ) = 119891 ((119909120582)Λ) forall (119909120582)Λ isin 119883 (91)



(119891) (92)





11988012058211205831205821


we have


(119891) = prod120582isinΛ

119882119872119881119880120582120583120582(119891120582) (93)





(⨂120582isinΛ

120583120582)((prod120582isinΛ


(94)

where


(95)








each (119899119898) isin N2

sup119909isin119883

119901 (119891119899 minus 119891119898) lt 120598 (96)


120583 (119891119899 minus 119891119898)) le 120598 (97)




120583 (1198911015840119899)



11989110158401015840119899 =

11989110158401198992 if 119899 is even

119891(119899minus1)2 if 119899 is odd (98)



2119899+1)119899isinN and(1198911015840119899)119899isinN = (11989110158401015840

2119899+1)119899isinN we get

1199061015840 = lim119899rarr+infin

WMV119880120583 (11989110158401015840

119899 )= lim


120583 (119891119899) = 119906lim


120583 (1198911015840119899) = lim


120583 (11989110158401015840119899 ) = 119906

(99)



WMV119880120583 (119891) = lim


120583 (119891119899) (100)


of Proposition 18



2 119899= 119898 997904rArr 119906119899 = 119906119898 (101)




2 119899 = 119898 997904rArr 119906119899 = 119906119898 (102)



2 (1 le 119899 lt 119898 le 119896) 997904rArr (119906119899 = 119906119898) (103)


120583otimes119899 (119880119899119896 ) = 120583otimes119899 ((1199091 119909119899)



WMV119880120583 (119891) =WMV119880Ord(119891)

120583otimesOrd(119891) (119891) (105)









119891 (119909) = 119891 ∘ 120587 (119909) forall119909 isin 119865 (107)




119891119899 ∘ 120587119898 = 119891119899 forall119899 le 119898 (108)




119865119891119899+1 = 119865119891119899 + 119865 (109)


This ends the proof


(prod120582isinΛ

119880120582119899) minusDΛ119899119909 = 0 (111)


WMV120582 (119891) (112)





119882119872119881120582 (119891120572) = 119882119872119881120582 (119891) (113)


WMV120582 ((119891119899)120572) =WMV120572minus119898120582 (119891119899) =WMV120582 (119891119899) (114)


119882119872119881120582 (119891120572) = 119882119872119881120582 (119891) (115)


119882119872119881120582 (119891V) = 119882119872119881120582 (119891) (116)



WMV120582 ((119891119899)V119899)= lim


=WMV120582 (119891) (117)


119882119872119881120582 (119891 ∘ 119906) = 119882119872119881120582 (119891) (118)






⨁119899isinN

119865119891119899 sub 119866 (120)


lim119899rarr+infin

119892119899 = 119892 (121)



WMV120582 (119892) =WMV120582 (119891) (123)


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11989110158401015840119899 =

11989110158401198992 if 119899 is even

119891(119899minus1)2 if 119899 is odd (98)



2119899+1)119899isinN and(1198911015840119899)119899isinN = (11989110158401015840

2119899+1)119899isinN we get

1199061015840 = lim119899rarr+infin

WMV119880120583 (11989110158401015840

119899 )= lim


120583 (119891119899) = 119906lim


120583 (1198911015840119899) = lim


120583 (11989110158401015840119899 ) = 119906

(99)



WMV119880120583 (119891) = lim


120583 (119891119899) (100)


of Proposition 18



2 119899= 119898 997904rArr 119906119899 = 119906119898 (101)




2 119899 = 119898 997904rArr 119906119899 = 119906119898 (102)



2 (1 le 119899 lt 119898 le 119896) 997904rArr (119906119899 = 119906119898) (103)


120583otimes119899 (119880119899119896 ) = 120583otimes119899 ((1199091 119909119899)



WMV119880120583 (119891) =WMV119880Ord(119891)

120583otimesOrd(119891) (119891) (105)









119891 (119909) = 119891 ∘ 120587 (119909) forall119909 isin 119865 (107)




119891119899 ∘ 120587119898 = 119891119899 forall119899 le 119898 (108)




119865119891119899+1 = 119865119891119899 + 119865 (109)


This ends the proof


(prod120582isinΛ

119880120582119899) minusDΛ119899119909 = 0 (111)


WMV120582 (119891) (112)





119882119872119881120582 (119891120572) = 119882119872119881120582 (119891) (113)


WMV120582 ((119891119899)120572) =WMV120572minus119898120582 (119891119899) =WMV120582 (119891119899) (114)


119882119872119881120582 (119891120572) = 119882119872119881120582 (119891) (115)


119882119872119881120582 (119891V) = 119882119872119881120582 (119891) (116)



WMV120582 ((119891119899)V119899)= lim


=WMV120582 (119891) (117)


119882119872119881120582 (119891 ∘ 119906) = 119882119872119881120582 (119891) (118)






⨁119899isinN

119865119891119899 sub 119866 (120)


lim119899rarr+infin

119892119899 = 119892 (121)



WMV120582 (119892) =WMV120582 (119891) (123)


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119891119899 ∘ 120587119898 = 119891119899 forall119899 le 119898 (108)




119865119891119899+1 = 119865119891119899 + 119865 (109)


This ends the proof


(prod120582isinΛ

119880120582119899) minusDΛ119899119909 = 0 (111)


WMV120582 (119891) (112)





119882119872119881120582 (119891120572) = 119882119872119881120582 (119891) (113)


WMV120582 ((119891119899)120572) =WMV120572minus119898120582 (119891119899) =WMV120582 (119891119899) (114)


119882119872119881120582 (119891120572) = 119882119872119881120582 (119891) (115)


119882119872119881120582 (119891V) = 119882119872119881120582 (119891) (116)



WMV120582 ((119891119899)V119899)= lim


=WMV120582 (119891) (117)


119882119872119881120582 (119891 ∘ 119906) = 119882119872119881120582 (119891) (118)






⨁119899isinN

119865119891119899 sub 119866 (120)


lim119899rarr+infin

119892119899 = 119892 (121)



WMV120582 (119892) =WMV120582 (119891) (123)


Competing Interests


References





























Volume 2014




Journal of











Journal of


Function Spaces






Algebra











⨁119899isinN

119865119891119899 sub 119866 (120)


lim119899rarr+infin

119892119899 = 119892 (121)



WMV120582 (119892) =WMV120582 (119891) (123)


Competing Interests


References





























Volume 2014




Journal of











Journal of


Function Spaces






Algebra
















Volume 2014




Journal of











Journal of


Function Spaces






Algebra









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