Absolutely continuous Banach space valued function

Question

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!

Answer 1

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.

Answer 2

Afaik, the right replacement for the concept of absolute continuity when dealing with Banach valued functions is the concept of continuous local primitives. It is not that surprising, because the $F$ is called a local primitive of $f$ if $$ \lim _{h \rightarrow 0} \int_a^b\left|\frac{1}{h}[F(x+h)-F(x)]-f(x)\right| d x=0 $$

For more context on this you can look at the example on page 172 of the book "The Bochner integral" by Mikusinski (1978), and at theorem Theorem 3.4 of the chapter XIII.