Nowhere Weak Differentiability of the Pettis Integral

For an arbitrary in nite-dimensional Banach space X, we construct examples of strongly-measurable X-valued Pettis integrable functions whose indefinite Pettis integrals are nowhere weakly di erentiable; thus, for these functions the Lebesgue Di erentiation Theorem fails rather spectacularly. We also relate the degree of nondi erentiability of the inde nite Pettis integral to the cotype of X, from which it follows that our examples are reasonably sharp. There are several generalizations of the space L1(R) of Lebesgue integrable functions taking values in the real numbers R (and de ned on the usual Lebesgue measure space ( ; ; ) on [0; 1] ) to a space of strongly-measurable \integrable" (suitably formulated) functions taking values in a Banach space X. The most common generalization is the space L1(X) of Bochner-Lebesgue integrable functions. Using the fact [P1, Theorem 1.1] that a strongly-measurable function is essentially separably-valued, one can easily extend Lebesgue's Di erentiation Theorem from L1(R) to L1(X). Speci cally [B; cf. DU, Theorem II.2.9], if f 2 L1(X), then lim h!0 1 h Z t+h t kf(!) f(t)k d (!) = 0 and so lim h!0 1 h Z t+h t f(!) d (!) = f(t) for almost all t in . Another generalization of L1(R) is the space P1(X) of strongly-measurable Pettis integrable functions. A function f : ! X is Pettis integrable if for each E 2 there is an element xE 2 X satisfying x (xE) = Z