Questions about Bochner integral I was wondering 


*

*If there is distinction between
existence of Bochner integral and
Bochner integrability, or the two always mean the same?

*If in Bochner integral, the
integrand is assumed to be
measurable wrt the Borel
$\sigma$-algebra of the codomain
Banach space?

*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?
 A: 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.
A: About 3. we have something even better! Hille's theorem.
http://fa.its.tudelft.nl/~neerven/publications/papers/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.
A: 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.
