Questions about Bochner integral

Question

I was wondering

1. If there is distinction between existence of Bochner integral and Bochner integrability, or the two always mean the same?
2. If in Bochner integral, the integrand is assumed to be measurable wrt the Borel $\sigma$-algebra of the codomain Banach space?

3. if differentiation under integral sign is still true for Bochner integral? What is the condition for that to be true? What kinds of derivatives are involved above, Fréchet derivative, the Gâteaux derivative, or something else?

   For example, $\frac{d}{dt} \int_a^t f(t) g(x,t) dt$, where $f: \mathbb{R} \rightarrow \mathbb{R}$, $g: B \times \mathbb{R} \rightarrow \mathbb{R}$, $B$ is a Banach space, and the Bochner integral exists.

Thanks and regards! Also are there some nice references?

Answer 1

About 3. we have something even better! Hille's theorem.

https://fa.ewi.tudelft.nl/~neerven/publications/notes/ISEM.pdf Theorem 1.19

Theorem 1.19 (Hille). Let $f : A \to E$ be $\mu$-Bochner integrable and let $T$ be a closed linear operator with domain $D(T)$ in $E$ taking values in a Banach space $F$ . Assume that $f$ takes its values in $D(T)$ $\mu$-almost everywhere and the $\mu$-almost everywhere defined function $T f : A \to F$ is $\mu$-Bochner integrable. Then $$T \int_A f \, d\mu = \int_A T f \, d\mu \,.$$

A lovely theorem. Your other questions can be answered by the first chapter of the refered document.

Answer 2

Anton Deitmar and coauthor(s) have recently been writing some things about this: e.g., http://arxiv.org/abs/1102.1246 Presumably they give references.

Answer 3

Here is some information:
http://en.wikipedia.org/wiki/Bochner_integral
A Bochner integrable function is almost all its values in a separable subspace, and (if the domain sigma-algebra is complete) is therefore measurable in your sense.