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?

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

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.

• This looks like a really nice set of notes. What is ISEM?
– t.b.
Commented Jun 28, 2011 at 23:26
• @Theo I nternet Sem inar. fa.its.tudelft.nl/isemwiki/moin.cgi/TitlePage Commented Jun 29, 2011 at 0:33
• @Theo: Internet Seminar as Willie says. Every year there is an internet seminar where you get lectures via internet in the first phase about something related to evolution equations, in the second phase there is a project with other participants and in the last phase there is a workshop in Blaubeuren. The ISEMs of this year and next year seem a bit boring to me... But they had nice things about SPDEs or Ergodic Theory.
– user1120
Commented Jun 29, 2011 at 10:02
• The link doesn't seem to work - is there another link? Commented Mar 20, 2019 at 15:08
• @Jeremy It's been a while since your comment, but just in case you were still interested: fa.its.tudelft.nl/~neerven/publications/notes/ISEM.pdf Commented May 7, 2020 at 20:59

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.

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.