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Let $X$ be a Banach space and $F:[a,b] \to X$ be an absolutely continuous function. Is it true that $F$ is differentiable almost everywhere? In particular, for any $f \in L^1([a,b],X)$, is the function $$ F(x) = \int_{[a,x]}f(t) dt $$ differentiable almost everywhere and $F'(x) = f(x)$ for almost all $x \in [a,b]$? (where the integral is the Lebesgue integral for Banach spaces)

I would also appreciate any literature where this is covered!

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  • $\begingroup$ Not certain what you mean by "the Lebesgue integral for Banach spaces". Are you referring to the Bochner Integral for vector measure spaces? You may want to check out Vector Measures by Diestel and Uhl. $\endgroup$ – Theo Bendit Mar 21 at 3:56
  • $\begingroup$ @Theo Bendit Yes, I did not know that it had a name. But it is defined the same as the usual Lebesgue integral for simple functions but the scalars are elements in a vector space. $\endgroup$ – Andrei Kh Mar 21 at 4:03
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This is not true in arbitrary Banach spaces, but for spaces $X$ possessing the "Radon-Nikodym property". A good resource might be the book "Vector measures" by Diestel and Uhl.

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