L∞ Norm Estimates for Conditional ExpectationsAuthor(s): Alan LambertSource: Proceedings of the Royal Irish Academy. Section A: Mathematical and PhysicalSciences, Vol. 97A, No. 2 (Dec., 1997), pp. 169-173Published by: Royal Irish AcademyStable URL: http://www.jstor.org/stable/20490245 .

Accessed: 12/06/2014 11:31

Your use of the JSTOR archive indicates your acceptance of the Terms & Conditions of Use, available at .http://www.jstor.org/page/info/about/policies/terms.jsp

.JSTOR is a not-for-profit service that helps scholars, researchers, and students discover, use, and build upon a wide range ofcontent in a trusted digital archive. We use information technology and tools to increase productivity and facilitate new formsof scholarship. For more information about JSTOR, please contact support@jstor.org.

.

Royal Irish Academy is collaborating with JSTOR to digitize, preserve and extend access to Proceedings of theRoyal Irish Academy. Section A: Mathematical and Physical Sciences.

http://www.jstor.org

This content downloaded from 185.44.78.76 on Thu, 12 Jun 2014 11:31:43 AMAll use subject to JSTOR Terms and Conditions

LX NORM ESTIMATES FOR CONDITIONAL EXPECTATIONS

By Alan Lambert

Department of Mathematics, University of North Carolina at Charlotte

[Received 18 September 1995. Read 30 October 1997. Published 1997.]

ABSTRACT

For a given probability space (X, F, p) and a subsigma algebra sl, we establish the existence of a convex subset C of the unit ball in L1(X, #, p) such that for any non negative F-measurable f,

1tE(fKs/)Koo=inft f

g }

Let (X, Y, u) be a complete probability space, and let si be a (complete) sigma subalgebra of F. All function and set statements encountered in this note are to be

interpreted as being valid up to sets of p-measure 0, and all functions and sets

encountered are assumed to be F-measurable. For a given sigma finite measure space

(Y, C, y), define W, to be the collection of sets in W of finite positive measure. This note

is concerned with developing formulas for calculating the essential supremum of the conditional expectation of a non-negative function. With this in mind we now list the

properties of conditional expectation pertinent to this investigation ([6] provides an in-depth source for information on conditional expectations).

We shall use the notation E'd for the conditional expectation operator: Elf =

E(f( si), defined for all non-negative f as well as fin any LP space, 1 < p < oo. Since

conditional expectation depends on the measure as well as the sigma algebra, when

more than one measure is being considered we will use notation such as E134 to

indicate the measure dependence. The symbol Ed' will be reserved for the 'original' measure p.

(i) For the sigma algebra sl a Y, Edf is the unique si-measurable function for

which fAf= fA Ed-lf for all A Es .

(ii) f> 0 a.e. => Edf 0 a.e.; f> 0 and f> 0 on a set of positive measure

Elf > 0 on a set of positive measure.

(iii) If a is s-measurable, then Ed (af) = a Edflf. There are many formulas for the essential supremum, Ilf II'), of a functionf. Our

starting point will be the following two forms:

(a <(A)

l(A)TA (b) If K100 = inf{r > 0: ,u(If 1 [(r, so)]

= 0}.

Note that if v is any sigma-finite measure mutually absolutely continuous with respect to p, then, as (b) above clearly shows, lf I,,) is the same whether ,u or v is used.

Proceedings of the Royal Irish Academy, Vol. 97A, No. 2, 169-173 (1997) ?D Royal Irish Academy

This content downloaded from 185.44.78.76 on Thu, 12 Jun 2014 11:31:43 AMAll use subject to JSTOR Terms and Conditions

170 Proceedings of the Royal Irish Academy

Lemma 1. Let f be non-negative and g strictly positive. Then

(c)f = supfAdl g o 00 < ,(A) fA gd/I

PROOF. Let dv = gdu. Then

= sup 1 - dv Igll 0 <v(A)v(A) Ag

= sup fAf/gdV 0 < v(A) lAdv

= SUp IAfdy < v(A)fA gdfl

= sup J 0 0 <p(A) IA gdy

The usual ways one evaluates the essential infimum of a function f is either to replace 'sup' by 'inf' in (a) or work with 1 /f. Lemma 1 leads to the vaguely symmetric

formulation:

- =sup y(A) I 0<#looO c1(A)fA gdPi

We now investigate essential supremums in connection with conditional expectations. It is fairly easy to switch from one conditional expectation to another when mutually absolutely continuous finite measures x and ,B are in use:

14' E~(d/3/da f) (d) E4'f= Ef(dfl/dod)

(This seems to be a well known result, but not readily found in the literature. A special

case of it is proven in [3], and the proof there easily generalises to (d).)

Remark. Since E~'(dfl/dac) is s-measurable, so is 1/E7'(dfl/dxc) and so (d) may be rewritten as

F/f - E(! d3/ da 4

= dfl/dcExrdflda

Now the function w E(dfl /da) satisfies the equation E w = 1 a.e. The collection

of such functions w plays a central role in Proposition 3, below. This collection of

functions has been used in other conditional expectation-related investigations (see, for example [1], [2] and [4]).

This content downloaded from 185.44.78.76 on Thu, 12 Jun 2014 11:31:43 AMAll use subject to JSTOR Terms and Conditions

LAMBERT-Estimatesfor conditional expectations 171

Consider f and g as in Lemma 1. Then E4flf > 0 and E I g > 0. Since these are

both s-measurable, we have

(e) Ed5f - s sup . Ed cc O <p(A);Ae-st fA E-g d4a

= sup fAfdP

O< < (A);Aed fA g df

Proposition 2. Let f > 0 and g > 0. The mapping y defined on the collection of all

subsigma algebras of YF given by y(Q) -= Eslfl is increasing and continuous with

respect to increasing sequences: E g

E 4f E"f - c J4 =- [Edglo

<1 IE-4g 0

and if {4} is an increasing sequence of sigma algebras generating the sigma algebra si, and iff and g are in L1, then

lmEd4nf _Ed4f

n:2>oo Esng cc cIE slc

PROOF. The order-preserving property of y follows directly from formula (e), since the larger the algebra, the larger the collection of sets over which to take the supremum.

For the increasing order continuity of y, note that

r2=limit w Eslc c

exists. Then, again by the order-preserving nature of y,

We may thus suppose that r is finite. Now suppose that

E-'f >r+c forsome e>0,

and let

s = {E > r+e/2}

Then ,u(S) > 0. Now by the a.e. convergence form of the increasing martingale convergence theorem ([5 p. 30]), Eshf-? E-4f a.e. and E4ng-*EEg a.e.

Moreover, ES g > 0 a.e. Thus

Ed4nf E-'lf

Ed4ng Edeg

It follows that for some positive integer n,

This content downloaded from 185.44.78.76 on Thu, 12 Jun 2014 11:31:43 AMAll use subject to JSTOR Terms and Conditions

172 Proceedings of the Royal Irish Academy

E{2 n }r+/4}>O.

But then

r >, E

;t lf r + e/4,

which is impossible. E

The order continuity fails for decreasing sequences of sigma algebras. For

example, let Xadmit a measurable partition into sets {A1, A2, . . .} with 4u(An) = 2-'. Let

4 be the purely atomic sigma algebra generated by the partition {A1 u ...

U A n+I,

An+2 ... Then sA D 4 za *, and n 4 = Y, the trivial sigma algebra consisting of

sets of measure 0 or 1 only. Definef to be 2 on AX for n odd, and 1 on An for n even.

Also, let g be similarly defined with the is and 2s reversed. Now, for any h, Ef h =

Jxh. Hence

Egf= f 2 2 1+D +L9 = 5/3,

x n=o n=l while

r X Io 00

E5g = = t-o2n+l+2 = 4/3. x n=o2 n=1l

Thus F7|tl - 5/4. But for each n, and any h,

Ed"n h =&(I1 A) h)XA U ...U A?ZQ(X)fh)xA. G(Al U ..U An). JI U .. A X1U A,A l((j JA n)1 j

In particular ||En Epf = 2. Indeed, by monotonicity, lim E In particlar Esin

j>n

sup g

= 2. Ednf

exists, is at least 2, and in particular is not 5/4. If we let Y be the trivial sigma algebra (consisting of sets of g measure 0 or I only),

then Eh = fxhdpu, and since Y a a Y, we see that

Jfxfdp- EJ4'f f4

fExg d | E-4 9 | g |

The final result in this note shows that IE'f[ II,, may be obtained by considering

If h Allo as h runs over a prescribed set of functions not depending onf. This may be viewed as a special L", LX form of a common L2, L' phenomenon. Indeed, if P is any

orthogonal projection on the Hilbert space L2, then for any fe L2:

PW112 = sup{l <Pf,g >: lgll2 <} 1

= sup{l<f,Pg>j: llgll2 < 1} = sup{J<fg>J: lIhIL2 < 1, hePL2}

= sup{ilfhhll: 1hAiL2 < 1, hePL2}

This content downloaded from 185.44.78.76 on Thu, 12 Jun 2014 11:31:43 AMAll use subject to JSTOR Terms and Conditions

LAMBERT-Estimates for conditional expectations 173

Proposition 3. Let C1) = {g > 0: EHlg = 1 a.e.}. Then for any f? 0, D1EdfIK = inf II lfg 11 : g c-(4)j}

PROOF. Suppose first thatf > 0, and let ge (s/). Then via the rightmost inequality in (f),

11 E 4f 11 oo- |E- gIf f g

Now let g =f/Edflf, which is in %(d). For this choice of g we have

g f 1f1 lE-f flY This shows that forf > 0

II EdflI II,, = minimum{ L :g ( )}4.

Now assume only thatf > 0. The result holds trivially if Edff L?, so assume that

Efllfe LI, and let N = {f = 0}. For t > 0, defineft =f+ tYN. Thenf, > 0,f1 > f, and

IIE ftIfAKo = minimum{ f| : ge()}

Also, for gc f (-d), ft/g - f/g so

inf{ fi :ge c d()} inf{ f' }:

-IIE-4filoo.

But Edft = Edf+ t -E XN, which converges, as t -> O, in L to Ef. This guarantees that

inf{ f : gec(KV)} ? -IIE-4fIIK,

which completes the proof. *

References

[1] Douglas, R. G. 1965 Contractive projections on an Lrspace. Pacifie Journal of Mathematics 15,

443-62.

[2] Dodds, P. G., Huijsmans, C. B. and De Pagter, B. 1990 Characterizations of conditional expectation

type operators. Pacific Journal of Mathematics 141 (1), 55-77.

[3] Lambert, A. and Weinstock, B. M. 1995 Descriptions of conditional expectations induced by non

measure preserving transformations. Proceedings of the American Mathematical Society 123 (3), 897-903.

[4] May, S.-T. 1954 Characterizations of conditional expectations as a transformation on function

spaces. Pacific Journal of Mathematics 4, 47-64.

[5] Parry, W. 1981 Topics in ergodic theory. Cambridge tracts in mathematics 75. London. Cambridge

University Press.

[6] Rao, M. M. 1993 Conditional measures and applications. New York. Marcel Dekker.

This content downloaded from 185.44.78.76 on Thu, 12 Jun 2014 11:31:43 AMAll use subject to JSTOR Terms and Conditions