DILWORTH and GIRARDI

For an arbitrary innnite-dimensional Banach space X, we construct examples of strongly-measurable X-valued Pettis integrable functions whose indefinite Pettis integrals are nowhere weakly diierentiable; thus, for these functions the Lebesgue Diierentiation Theorem fails rather spectacularly. We also relate the degree of nondiierentiability of the indeenite Pettis integral to the cotype of X, from which it follows that our examples are reasonably sharp. There are several generalizations of the space L 1 (R) of Lebesgue integrable functions taking values in the real numbers R (and deened 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 L 1 (X) of Bochner-Lebesgue in-tegrable functions. Using the fact P1, Theorem 1.1] that a strongly-measurable function is essentially separably-valued, one can easily extend Lebesgue's Diieren-tiation Theorem from L 1 (R) to L 1 (X). Speciically B; cf. DU, Theorem II.2.9], if f 2 L 1 (X), then lim h!0 1 h Z t+h t kf(!) ? f(t)k dd(!) = 0 and so lim h!0 1 h Z t+h t f(!) dd(!) = f(t) for almost all t in .