Absolutely continuous and differentiable almost everywhere

Question

I've read the following claim and I wonder if someone can direct me to or provide me with a proof of it:

"A strongly absolutely continuous function which is differentiable almost everywhere is the indefinite integral of strongly integrable derivative"

It was in the context of Bochner integrable functions so I'm assuming that "strongly" means with respect to the norm.

Thanks!

Answer 1

An absolute continuous function $f:[a,b] \to X$ is differentiable almost everywhere such that for a fixed $y \in [a,b]$

$$f(x) = \int_x^y f'(z) \, \text{d} z + f(y) \quad \text{for all } x \in [a,b].$$

In particular $f'$ is strongly integrable, i.e. $f' \in L^1(a,b;X)$.

You can find this in the book "Measure Theory and Fine Properties of Functions" by Evans and Gariepy. Also in the book "Linear Functional Analysis" by Alt.