1
$\begingroup$

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!

$\endgroup$
0
$\begingroup$

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.

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.