Pacific Journal ofMathematics

GAUSSIAN NULL SETS AND DIFFERENTIABILITY OFLIPSCHITZ MAP ON BANACH SPACES

ROBERT RALPH PHELPS

Vol. 77, No. 2 February 1978

PACIFIC JOURNAL OF MATHEMATICSVol. 77, No. 2, 1978

GAUSSIAN NULL SETS AND DIFFERENTIABILITYOF LIPSCHITZ MAP ON BANACH SPACES

R. R. PHELPS

In this note we introduce and briefly study the notion ofa Gaussian null set in a real separable Banach space E.As a corollary to recent work of Aronszajn we then showthat a locally Lipschitz mapping from E into a Banachspace with the Radon-Nikodym property is Gateaux differen-tiable outside of a Gaussian null set. This is an infinitedimensional generalization of Rademacher's classical theoremthat such mappings from Rn to Rm are differentiate almosteverywhere (Lebesgue). This approach will be comparedwith another generalization of Rademacher's theorem dueindependently to Christensen and Kaier and to Mankiewicz.

In order to prove an extension of Rademacher's theorem, onefirst needs a generalization of the notion of a set of Lebesguemeasure zero. Since we are interested in a question involving con-tinuous functions, we restrict our attention to Borel sets. Whatwe want to define, then, is a class of Borel sets (which will even-tually be considered as the class of null sets) which is closed withrespect to countable unions and translations. Moreover, we wanta Borel subset of a null set to be itself a null set, and we wantour class to coincide with the Borel sets of Lebesgue measure zeroin finite dimensional spaces. We also require that a nonempty openset not be a null set. Now, it is well known that there is noanalogue to Lebesgue measure in infinite dimensional spaces; in fact,there does not even exist a positive σ-finite measure on l2 whosenull sets are translation invariant [14, p. 108]. Thus, we cannotsimply use the class of null sets of some fixed measure. We can,however, use the common null sets of a family of measures, namely,the family of nondegenerate Gaussian measures.

DEFINITION. A nondegenerate Gaussian measure μ on the realline R is one having the form

( * ) μ(B) = (2πb)~1/2\ exp [-(2&)"1(ί - a)2]dt

where B is a Borel subset of R and the constant b is positive. Thepoint a 6 R is called the mean of μ.

Such measures are obviously mutually absolutely continuouswith respect to Lebesgue measure (on the Borel sets) and the pro-

523

524 R. R. PHELPS

duct of n such measures is equivalent to Lebesgue measure on Rn.As we will see, any nondegenerate Gaussian measure on E is (essen-tially) a countable product of such measures. First, we recall thefollowing definition.

DEFINITION. A probability measure λ on the Borel subsets ofthe real Banach space E is said to be a nondegenerate Gaussianmeasure of mean xQeE if for each feE*, / Φ 0, the measureμ = λo/"1 has the form (*) (above), where a — f(x0).

The lemma which follows is proved by combining a number ofknown results, but does not itself appear to be stated explicitly inthe literature.

LEMMA 1. Suppose that μ is a nondegenerate Gaussian measureof mean 0 on the separable infinite dimensional Banach space E.Then there exists a sequence {en} £ E and a one-to-one continuouslinear map T: E —> RN {the countable product of lines with the pro-duct topology) with the following properties:

( i ) The linear span E^ of {en} is dense in E.(ii) For each n, the image of en under T is δn9 the sequence

having 1 in the nth. place, 0 elsewhere.(iii) For each Borel subset B Q E, the set TB is a Borel subset

of RN.(iv) The measure v = μoT~ι is a countable product of non-

degenerate Gaussian measures vn on R of mean 0.

Proof. Kuelbs [8, 9] has shown the following: Let {xn} bedense in E, choose {/„} £ E* such that fn(xn) = \\xn\\ and | |/w | | = 1for each n and define

(x, y) = Σ2-nfn(x)fn(y) , x , y e E .

Then this defines an inner product on E with associated normIMI2 = (Xf aθ1/2 ̂ IMI The completion of (E, \\.. ||2) is a separableHubert space H and the natural embedding TX\E —> H carries Borelsubsets of E into Borel subsets of H. Moreover, the density ofTγE in H implies that μt — μ°Tΐι is a nondegenerate Gaussianmeasure on H. From Chapter 1 of Skorohod [14] it follows thatthere exists an orthonormal basis {en} for H and a sequence {vn} ofnondegenerate Gaussian measures on the line with the followingproperties. First, the natural continuous linear embedding T2:H-+RN [defined by T2(Σtnen) = (ίΛ)] maps Borel subsets of H into Borelsubsets of RN. [The basis {en} is the sequence of normalized eigen-vectors for the strictly positive nuclear correlation operator associa-

GAUSSIAN NULL SETS AND DIFFERENTIABILITY 525

ted with μιm] Second, if v = Πvn, then v = μ^ Ti1; this makes sense,since RN is separable and metrizable, so the Borel sets coincidewith the usual product space σ-algebra generated by the cylindersets based on Borel sets. Now, Kuelbs [8] has shown that thesequence {en} is actually contained in TJΞ, i.e., in E. Moreover, hehas proved that given feE* there exists for each n a Borel measur-able function Pn, defined ^-almost everywhere on E with values inspan {elf 9en}, such that the sequence f°Pn converges μ-almosteverywhere to /. Suppose that the span E of {en} were not densein E; then there would exist feE*, f Φ 0, such that f(EJ) = 0. Inparticular, this would imply that /<>Pn = 0 a.e. μ and hence that/ = 0 a.e.μ, contradicting the fact that μ°f~ι is nondegenerate onR. This proves part (i). To prove the remaining parts we simplylet Γ = T2oT,.

DEFINITION. A Borel subset B of the separable Banach spaceE will be called a Gaussian null set if μ(B) = 0 for every nondegen-erate Gaussian measure μ on E. The family of all Gaussian nullsets will be denoted by 5f.

PROPOSITION 2. The family 5f of Gaussian null sets has thefollowing properties:

( i ) The countable union of members of & is an element of& and a Borel subset of a member of ^ is in ^.

(ii) For all B e & and xeE, the translate x + B is in &.(iii) If U £ E is open and nonempty, then U&&.(iv) If S: E —> E is an isomorphism (that is, one-one, linear,

continuous and onto), then S(B)e& for every Be^.(v) If E is finite dimensional (hence isomorphic to Rn for

some n), then a Borel set B is in ^f if and only if B has Lebesguemeasure zero.

Proof. Part (i) is immediate from the definition. Part (ii)follows from the easily verified fact that a translate of a nondege-nerate Gaussian measure is again such a measure (not necessarilyabsolutely continuous with respect to the original [10]). Part (iii)is shown in Corollary 4 below. Part (iv) is a consequence of theobservation that if μ is a nondegenerate Gaussian measure, then sois the measure μ<>S; this fact is immediate from the definition.Finally, part (v) is proved by noting [14] that a Gaussian measureon a finite dimensional space is mutually absolutely continuous withLebesgue measure.

LEMMA 3. // the sequence {wn} £ E has dense linear span and

526 R. R. PHELPS

satisfies | |ww | |-^0, then its symmetric closed convex hull K iscompact and is not a Gaussian null set.

Proof. Define L: l2 —• E by setting, for x = (xn) el2,

Lx = Σ2~nxnwn .

It is clear that L is linear and has dense range. Let U denote theunit ball of l2. If x e U, then | xn | <; 1 for all n, hence Lx e K.Since K is the closed convex hull of the compact set {±wn} U {0}, itis compact. In particular, it is bounded, which shows that T iscontinuous. It follows from the definition that if μ is any nonde-generate Gaussian measure on lZJ then λ = μoT'1 is a nondegenerateGaussian measure on E. Moreover, if μ(U)>0, then, since T^KΏ,U, we have \(K) = μiT^K) ^ μ(U) > 0. It is known (see, e.g.,[13]) that any nondegenerate Gaussian measure μ on l2 assignspositive measure to any nonempty open set, so K is not Gaussiannull.

COROLLARY 4. If U is any nonempty open subset of E, thenU is not a Gaussian null set.

Proof. Since E is separable, it can be expressed as a countableunion of translates of U, hence would itself be Gaussian null if Uwere. Thus, we merely need to show the existence of at least onenondegenerate Gaussian measure on E, and this is a consequenceof Lemma 3.

Note that the above result implies that the complement ofa Gaussian null set is dense in E.

We next define Aronszajn's class ό$? of exceptional sets andcompare them with the Gaussian null sets.

DEFINITION. Let {an} c E be a sequence of nonzero elementswhich has dense linear span in E. Define J^{an) to be the familyof all Borel sets of the form \JZ=1An, where each An is a Borel setwith the property that for each xeE, the set

(An + x)f] Ran

has Lebesgue measure zero in the line Ran. Finally, let Jzf be theintersection of the families J*f{an}, over all possible such sequences

Aronszajn [1] has shown that the family J ^ has all the proper-ties listed (for the Gaussian null sets) in Proposition 2.

GAUSSIAN NULL SETS AND DIFFERENTIABILITY 527

PROPOSITION 5. If a Borel subset A of E belongs to Aronszajn'sclass j y of exceptional sets, then it is a Gaussian null set.

Proof. Since J ^ is translation invariant it suffices to provethat μ(A) = 0 for each nondegenerate Gaussian measure μ of mean0 on E. Choose {en} Q E and T:E-^RN as in Lemma 1. By hypo-thesis, it is possible to write A = U An where each Borel set An

has the property that An Π (x + Ren) is Lebesgue null for all x e E.We need only show μ(An) = 0 for each n. By Lemma 1, B = TAn

is a Borel subset of RN and v = μ<> T~ι is a product of one dimen-sional nondegenerate Gaussian measures vn of mean 0. Thus, wewant v(B) — 0. Since T is one-one, we have T(An Π (x + Ren)) = Bf](Tx + Rdn). The latter has Lebesgue measure zero in the line Rδ%,since T maps Ren linearly onto Rdn. Thus, B Γ) (y + Rδn) is Lebes-gue null for all yeTE. Let g=χB By the pointwise Fubini-Jessentheorem [7, p. 209], v(B) = \gdv = lim^.^ gk(x) for v-almost all xeRM,where, for x = (xn) e RN,

= \ -dvk(yk) .

Since v{TE) = 1, we can fix xe Ti£ such that the above limit exists.It clearly suffices to show that gh(x) = 0 if k^ n. But for anysuch fc, the integrand appearing in the definition of gk(x) can beexpressed (using Fubini's theorem) as the integral with respect todvx x x ώv,! x dvn+ι x x dvk of

••-,!/*, , l/*f »*+» %+2, -)dvn(yn) .

The integrand in this last integral is the characteristic function ofthe set

B Π [(yιf , yw-!, 0, yn+1, , I/*, xfc+i, a?A+a, •) + RSn]

which is of the form B Π (s + Λ^J, where

- % , 0, 0, 0, . ..) + * .

Clearly, the finitely nonzero sequences are in TE and x e TE, soz 6 TE and therefore this set has one dimensional Lebesgue measurezero. It follows that gk(x) = 0 and the proof is complete.

We do not know whether &f is a proper subset of <&.

We now give the definitions which are needed to apply thenotion of Gaussian null sets to differentiability of Lipschitz mappings.

528 R. R. PHELPS

DEFINITIONS. Suppose that G is a nonempty open subset of thereal Banach space X and that T: G —> Y is a mapping from G intothe real Banach space Y. We say that T is Gateaux differentiateat a? e G if for each u e X the limit

( d ) lim T(x + tu) - T(x)tt-*0

exists in the norm topology of Y and defines a linear (in u) mapwhich is continuous from X to Y.

We say that T is locally Lipsehitz if for each x e G there existpositive constants M and δ such that

\\T(y)- T(z)\\ ^ M\\y - z\\

whenever y, zeG and \\y — x\\ < δ and ||g — x\\ < S.Finally, a real Banach space F is said to have the Radon-

Nikodym property (RNP) provided every function of boundedvariation from [0, 1] into Y is differentiate almost everywhere.

The name for this class of spaces (which contains separable dualspaces and reflexive spaces) arises from the fact that a Radon-Nikodym theorem is valid for vector measures with values in suchspaces. For convenience we have chosen as our definition one of alarge number of known characterizations of the RNP; see [5] or[6] for a comprehensive survey.

THEOREM 6. (Aronszajn) Suppose that X is a separable realBanach space, that Y is a real Banach space with the Radon-Nikodym property and that G is a nonempty open subset of X. IfT: (? —> Y is a locally Lipsehitz mapping, then T is Gateaux diffe-rentiate at all points x of a Borel subset of G whose complementis in jy\ In particular, it is Gateaux differentiate outside of aGaussian null subset of G.

This is essentially Theorem 1 of Chapter 2 of [1]. In the latter,the space Y was assumed to be a separable dual space or a reflexivespace (hence a space with the RNP). Since T(G) is separable, so isits closed linear span, hence one can assume that Y is separable.It is an open question of some standing whether a separable spacewith the RNP can be embedded in a separable dual space [5, 6].If the answer is affirmative, then Aronszajn's proof is obviouslyadequate as it stands. In any event, the hypothesis that the separa-ble space Y be a dual space is only used in two places in Chapter2 of [1]: In Lemma 1 it is used to show that a Lipsehitz map froma real interval into Y is differentiate almost everywhere, and this

GAUSSIAN NULL SETS AND DIFFERENTIABILITY 529

is immediate from our definition of the RNP. In Lemma 2 it isused to produce a dense sequence {bk} in the predual of Y, but theproof is valid if {bk} is merely a total sequence in Y*. With thesechanges in [1], then, the above theorem is valid as stated. (Wecaution the reader that our terminology differs from that of [1].For instance Aronszajn says T has a ''Gateaux differential at x" ifthe limit in (d) exists for all u and for positive t, with no assump-tion about linearity or continuity of the resulting map. Moreover,where we say "T is Gateaux differentiate at x"9 Aronszajn writes"DT(x) is a differential.")

A different class of Borel null sets has been defined by Chris-tensen [2, 4] and used by him and Kaier [3] and (independently) byMankiewicz [11,12] to prove differentiability theorems for Lipschitzmaps in Frechet spaces. Christensen calls a Borel subset B of anabelian Polish topological group G a Haar zero set if there existsa Borel probability measure μ on G such that μ(B + x) = 0 for allxeG. The family of all such sets is clearly closed under transla-tions. It is less obvious (but true) that it is closed under countableunions. Moreover, it is defined in any separable Frechet space andagrees with Lebesgue null Borel sets in finite dimensional spaces.In an infinite dimensional space every compact set is a Haar zeroset but (by Lemma 3) this is not true of the Gaussian null sets.On the other hand, it is immediate from the definitions that anyGaussian null set is a Haar zero set. Consequently, to say that amap is differentiable outside of a Haar zero set is weaker thansaying that it is differentiable outside of an Aronszajn exceptionalset.

The following example shows that the Gaussian null sets (andhence Aronszajn's exceptional sets) fail to have a useful propertypossessed by Lebesgue measurable sets. Recall that any pairwisedisjoint family of sets of positive Lebesgue measure is at mostcountable. Christensen [2] has posed the question as to whetherthe analogous property is valid for the Haar zero sets; this seemsstill to be open.1

EXAMPLE. There exists an uncountable collection of pairwisedisjoint compact subsets of l2, each of which is not a Gaussian nullset.

Proof. For each sequence s = (sj such that sn — ± 1 for eachn let

1 Added in proof. Christensen has informed us that Ryll-Nardjewski has constructeda simple example in the countable product of lines which shows that the answer isnegative.

530 R. R. PHELPS

K(s) = {(xn) G l2: \xn - sn2~n\ ^ A"\ n = 1, 2, 3, ...} .

Each K(s) is a compact convex Hubert cube centered at the pointc(s) = (sn2~~n). If s Φ. s\ then there exists an index m such thatl«« — s»| = 2. Consequently, if xGiΓ(s) n i£(s')> then

mΔ — S m £ Sm^5 j ^ I Xm Smώ \ -f I X m — S w £ I ^ ώ 4t ,

a contradiction which shows that the sets K(s) form a pairwisedisjoint family, which is clearly uncountable. To show that eachK(s) is not a Gaussian null set it suffices to show that each translate

K(s) - c(s) = {x e l2: \xn\ ^ 4~n , w = 1, 2, 3, •}

is not Gaussian null. If en denotes the canonical wth basis vectorin l2, then the sequence wn = 4~we% satisfies the hypothesis of Lemma3. Since its symmetric closed convex hull is contained in K(s) —c(s), the latter is not a Gaussian null set.

As has been noted in [1], there is no possibility of replacingGateaux differentiability in Theorem 6 by Frechet differentiability(where the limit in (d) is assumed to exist uniformly for \\u\\ <> 1).The simplest example seems to be the norm in l19 which is Gateauxdifferentiate at each point with all nonzero coordinates, but isnowhere Frechet differentiable. For a discussion of Gateaux diffe-rentiability of continuous convex functions (which are necessarilylocally Lipschitz) we refer the reader to [1], where it is shown thatfor such functions the conclusion to Theorem 6 can be strengtheneda bit.

J. Diestel has informed us that a modification of the discussionin [5, p. 107] shows that if E has the property that every Lipschitzmap from [1] into E is differentiable a.e., then E has the RNP.Thus, the latter class of spaces is the most general one for whichTheorem 6 remains valid. For a discussion of the relationshipbetween differentiability of Lipschitz functions and the isomorphicclassification of Banach spaces, see [5, p. 118].

We wish to thank Professors R.A. Blumenthal and James D.Kuelbs for conversations on material related to this paper.

REFERENCES

1. N. Aronszajn, Differentiability of Lipschitzian mappings between Banach spaces,Studia Math., 57 (1976), 147-190.2. J.P.R. Christensen, On sets of Haar measure zero in abelian Polish groups, IsraelJ. Math., 13 (1972), 255-260.3. f Measure theoretic zero sets in infinite dimensional spaces and applica-tions to differentiability of Lipschitz mappings, 2-ieme Coll. Anal. Fonct. (1973, Borde-

GAUSSIAN NULL SETS AND DIFFERENTIABILITY 531

aux), Publ. du Dept. Math. Lyon t. 10-2 (1973), 29-39.4. J. P. R. Christensen,, Topology and Borel Structure, Math. Studies 10, North-Holland/American Elsevier, New York, 1974.5. J. Diestel and J.J. Uhl, Jr., The theory of vector measures, Amer. Math. Soc.Surveys, Vol. 15 (1977).6. , The Radon-Nikodym theorem for Banach space valued vector measures,Rocky Mountain J. Math., 6 (1976), 1-46.7. N. Dunford and J. Schwartz, Linear Operators, Part I, Interscience, New York,(1958).8. J. Kuelbs, Gaussian measures on a Banach space, J. Functional Analysis, 5 (1970),354-367.9. , Expansions of vectors in a Banach space related to Gaussian measures,Proc. Amer. Math. Soc, 27 (1971), 364-370.10. H-H. Kuo, Gaussian measures in Banach spaces, Lecture Notes in Math. 463,Springer-Verlag, Berlin-Heidelberg-New York, 1975.11. P. Mankiewicz, On topological, Lipschitz, and uniform classification of LF-spaces,Studia Math., 52 (1974), 109-142.12. , On the differentiability of Lipschitz mappings in Frechet spaces, StudiaMath., 45 (1973), 15-29.13. B. S. Rajput, The support of Gaussian measures on Banach spaces, Theory Prob.and Appl., 17 (1972), 728-733.14. A.V. Skorohod, Integration in Hilbert space, Ergeb. der Math. u. ihrer Grenzgeb.vol. 79, Springer-Verlag, New York, 1974.

Received September 2, 1977. Work supported in part by a grant from the NationalScience Fundation.

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Pacific Journal of MathematicsVol. 77, No. 2 February, 1978

Graham Donald Allen, Duals of Lorentz spaces . . . . . . . . . . . . . . . . . . . . . . . . . . 287Gert Einar Torsten Almkvist, The number of nonfree components in the

decomposition of symmetric powers in characteristic p . . . . . . . . . . . . . . 293John J. Buoni and Bhushan L. Wadhwa, On joint numerical ranges . . . . . . . . 303Joseph Eugene Collison, Central moments for arithmetic functions . . . . . . . . 307Michael Walter Davis, Smooth G-manifolds as collections of fiber

bundles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 315Michael E. Detlefsen, Symmetric sublattices of a Noether lattice . . . . . . . . . . . 365David Downing, Surjectivity results for φ-accretive set-valued

mappings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 381David Allyn Drake and Dieter Jungnickel, Klingenberg structures and

partial designs. II. Regularity and uniformity . . . . . . . . . . . . . . . . . . . . . . . 389Edward George Effros and Jonathan Rosenberg, C∗-algebras with

approximately inner flip . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 417Burton I. Fein, Minimal splitting fields for group representations. II . . . . . . . . 445Benjamin Rigler Halpern, A general coincidence theory . . . . . . . . . . . . . . . . . . 451Masamitsu Mori, A vanishing theorem for the mod p Massey-Peterson

spectral sequence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 473John C. Oxtoby and Vidhu S. Prasad, Homeomorphic measures in the

Hilbert cube . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 483Michael Anthony Penna, On the geometry of combinatorial manifolds . . . . . 499Robert Ralph Phelps, Gaussian null sets and differentiability of Lipschitz

map on Banach spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 523Herbert Silverman, Evelyn Marie Silvia and D. N. Telage, Locally univalent

functions and coefficient distortions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 533Donald Curtis Taylor, The strong bidual of 0(K ) . . . . . . . . . . . . . . . . . . . . . . . . 541Willie Taylor, On the oscillatory and asymptotic behavior of solutions of

fifth order selfadjoint differential equations . . . . . . . . . . . . . . . . . . . . . . . . . 557Fu-Chien Tzung, Sufficient conditions for the set of Hausdorff

compactifications to be a lattice . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 565

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