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Non Differentiability of Pettis Primitives

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Non Differentiability of Pettis Primitives Non differentiability of Pettis primitives Udayan B. Darji University of Louisville Joint work with B. Bongiorno, L. Di Piazza May 17, 2012 Preliminaries 1 Throughout X denotes a Banach space unless otherwise indicated. k k denotes the norm of this Banach space. We consider functions from [0; 1] into X. We use the standard Lebesgue measure λ on [0; 1]. Let f : [0; 1] ! X. Function f is strongly measurable if there exists a sequence of simple functions ffng converging a.e. to f. Moreover, if ffng satisfies the additional property that Z limn!1 kf − fnkdλ = 0; [0;1] then we say that f is Bochner integrable. In such a case, we let Z Z (B) f = lim fn: [0;1] n!1 [0;1] Recall that (B) R f is independent of the choice of ffng. 2 As usual, we let X∗ denote the dual of X. We say that f : [0; 1] ! X is weakly measur- able, if x∗f is λ-measurable for all x∗ 2 X∗. We call a weakly measurable function f : [0; 1] ! X Pettis integrable if for each measurable set A ⊆ [0; 1] there is xA 2 X such that Z ∗ ∗ x (xA) = x fdλ, A for all x∗ 2 X∗. In the case that f is Pettis integrable, we use Z (P ) f [0;1] to denote the Pettis integral of f. 3 Basics Properties of Bochner Integral and Pettis Integrals 4 We next recall some basic well-known facts and differences between the Bochner integral and Pettis integral. Fact 1 For any X, f is Bochner integrable ) f is Pettis integrable: n Fact 2 For X finite dimensional (i.e., X = R ) we have the following: f is Lebesgue m f is Bochner integrable m f is Pettis integrable Moreover, values of R f under all of the above integrals coincide. 5 Fact 3 There are weakly measurable functions which are not strong measurable. Fact 4 The following are equivalent: • f is strongly mesurable. • f is weakly measurable and the range of f is essentially separable, i.e., there exists a set E of measure zero such that f([0; 1] n E) is separable. Corollary 5 If X is a separable Banach space, then strong measurability and weak measura- bility coincide. 6 Fact 6 Let f : [0; 1] ! X be strongly measur- able. Then, f is Bochner integrable m R kfk < 1 Fact 7 Let f : [0; 1] ! X be Bochner and F be its primitive, i.e., F is the X-valued measure on [0; 1] defined by Z (B) f = F (A): A Then, F is countably additive, λ-continuous and is of finite variation. Moreover, for almost every t 2 [0; 1], we have that F (t + h) − F (t) lim = f(t): h!0 h 7 Fact 8 Let f : [0; 1] ! X be Pettis and F be its primitive, i.e., F is the X-valued measure on [0; 1] defined by Z (P ) f = F (A): A Then, F is countably additive, λ-continuous and is of σ-finite variation. Theorem 9 (Dilworth-Girardi, 1995) Let X be any infinite dimensional Banach space. Then, there exists Pettis integrable f : [0; 1] ! X such that F , the primitive of f, is nowhere differentiable! More precisely, for every t 2 [0; 1] we have that F (t + h) − F (t) lim = 1: h!0 h 8 Motivation 9 It is generally considered that the class P of Pettis integrable functions is much larger than the class B of Bochner integrable functions whenever X is infinite dimensional. Indeed, if X is not separable then, the class of strongly measurable functions is much smaller than the class of weakly measurable functions as the range of strongly measurable functions is essentially separable. However, even when X is separable and infinite dimensional class P is considered much smaller than the class B. How does one make this statement precise? 10 A natural notion of smallness in a complete metric space is that of first category. The space B is a Banach space, however, the class P enjoys no natural complete metric. Hence, it is difficult to make a meaningful use of cat- egory in this setting. Moreover, given any Bochner primitive F 2 C([0; 1];X), arbitrarily close to it, in the norm of C([0; 1];X), there are Pettis primitives. However, both of B and P are vector spaces. Our investigation was motivated by the follow- ing question: How large a set A ⊂ P must be so that < B[A >= P? 11 A large subset of P n B 12 The concept of unconditional convergence is central to the investigation of Pettis integral. Let fxng be a sequence in X. We say that fxng is unconditionally convergent if every re- arrangement of fxng converges. If X is finite dimensional, then fxng is uncon- ditionally convergent iff fxng is absolutely con- vergent. Theorem 10 (Dvoretzky-Rogers) If X is infi- nite dimensional, then there is a sequence fxng in X which converges unconditionally but does not converge absolutely. 13 Fact 11 Let fEng be a partition of [0; 1] and let fxng be a sequence in X. P1 • n=1 xnχEn is Bochner integrable iff P1 n=1 kxnkλ(En). P1 P1 • n=1 xnχEn is Pettis integrable iff n=1 xnλ(En) is unconditionally convergent. P1 In particular, if n=1 xnλ(En) converges un- P1 conditionally but not absolutely, then n=1 xnχEn is Pettis integrable but not Bochner integrable. 14 Theorem 12 (BDDi, Main theorem) There ex- ists a subset K ⊂ P of the cardinality contin- uum such that the following holds: • Each f 2 K is nowhere differentiable. • If f1; : : : ; fn 2 K and λ1; : : : ; λn 2 R are not all zero, then n X λifi 2= B: i=1 15 Theorem 13 (BDDi, First Approximation) There exists a subset K ⊂ P of the cardinality contin- uum such that the following holds: • If f1; : : : ; fn 2 K and λ1; : : : ; λn 2 R are not all zero, then n X λifi 2= B: i=1 16 Theorem 14 (BDDi, Second Approximation) There exists a subset K ⊂ P of the cardinality continuum such that the following holds: • For each f 2 K and subinterval I of [0; 1] we have that fjI is not Bochner. • If f1; : : : ; fn 2 K and λ1; : : : ; λn 2 R are not all zero, then n X λifi 2= B: i=1 17 Proof of the First Approximation. Recall that there is a set a A of cardinality continuum such that each A 2 A is an infinite subset of N and if A; B are distinct elements of A, then A \ B is finite. (Such a collection is called an almost disjoint collection.) Let fxng be an unconditionally convergent se- quence X which does not converge absolutely. Let I1;I2;::: be a parition of [0; 1) into nonover- lapping intervals. For each A 2 A, we define a function fA 2 P n B in the following fashion. List A as a1; a2;::: in an increasing order. 18 Then, 1 1 f ≡ X x χ : A i Iai i=1 jIaij Let K = ffA : A 2 Ag. Clearly, fA 2 P n B. Let be distinct elements of and fA1; : : : fAn K λ1; : : : λn 2 R be non zeros. As the collection A is almost disjoint, there is n 2 A1 such that for (n; 1) \ A1 \ Ai = ;, for 2 ≤ i ≤ n. Then Pn the function i=1 λifi is equal to λ1f1 on Im for all m 2 A1, m > n. Hence, we have that Pn i=1 λifi is not in B. End of proof of the First Approximation. 19 Proof of the Second approximation. We pro- ceed with some notations and then lemmas. Let n1; n2;::: be a sequence of integers greater than 1. Then, (n1; n2;:::)-tree, denoted by T (n1; n2;:::), is defined as 1 [ i T (n1; n2;:::) = f;g Πj=1f0; : : : ; nj − 1g: i=1 If σ 2 T (n1; n2;:::), then jσj denotes the length of σ and if n ≤ jσj, then σjn denotes the restric- tion of σ to the first n terms. If σjn = τ, then we say that σ is an extension of τ or τ is a restriction of σ. If σ 2 T (n1; n2;:::) and i 2 N, then σi is a extension of σ of length n+1 whose n + 1 term is i. If σ 2 T (n1; n2;:::), then [σ] = fτ 2 T (n1; n2;:::): τ is an extension of σ:g 20 Associated with (n1; n2;:::), we also define an interval function. More specifically, with each σ 2 T (n1; n2;:::), we associate Iσ an subinter- val of [0; 1] such that the following holds. 1. For each k, fIσ : σ 2 T (n1; n2;:::); jσj = kg is partition of [0; 1] into n1 · nk non- overlapping closed intervals. 2. For each σ 2 T (n1; n2;:::) with jσj = k, we have that fIσl : 0 ≤ l ≤ nk+1 − 1g is a parti- tion of Iσ into non-overlapping intervals of equal length. 21 Lemma 15 There exists a function T : T (n1; n2;:::) ! X such that the following condition holds. For all σ 2 T (n1; n2;:::), and any bijection α : N ! [σ], the sequence fxng defined by xn = T (α(n)) converges unconditionally but not absolutely. Moreover, we can make the function T 1-1, if necessary. Lemma 16 Let fA g be a pairwise σ σ2T (n1;n2;:::) disjoint system of positive measure subsets of [0; 1] such that Aσ ⊆ Iσ and let T be a function of the Lemma 15.
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