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Nowhere Weak Differentiability of the Pettis Integral

Stephen J. Dilworth·Maria Girardi
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Abstract

For an arbitrary infinite-dimensional Banach space $\X$, we construct examples of strongly-measurable $\X$-valued Pettis integrable functions whose indefinite Pettis integrals are nowhere weakly differentiable; thus, for these functions the Lebesgue Differentiation Theorem fails rather spectacularly. We also relate the degree of nondifferentiability of the indefinite Pettis integral to the cotype of $\X$, from which it follows that our examples are reasonably sharp. This is an expanded version of a previously posted paper with the same name.

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Stephen J. Dilworth, Maria Girardi. 1994-10-31. Nowhere Weak Differentiability of the Pettis Integral. https://arxiv.org/abs/math/9410203

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