Let $f:X \to B$ be a Banach-valued map, where $X$ is a measure space. I am currently studying Bochner integration theory (that is, a theory for integrating Banach-valued functions).
In "Infinite Dimensional Analysis: A Hitcherhiker's guide", the authors give the following theorem, without proof.
If $X$ is a finite measure space and $f$ is measurable (that is, each Borel set has a measurable preimage), then $f$ is Bochner integrable if and only if $|f|$ is integrable in the usual sense.
In "Functional Analysis" by Kosaku Yosida, the following is given.
If $f$ is strongly measurable (a.e. limit of measurable simple functions), $f$ is Bochner integrable if and only if $|f|$ is integrable in the usual sense.
I am quite confused because of the slightly different settings. In the first theorem, $f$ is measurable while in the second theorem, $f$ is "strongly" measurable. (I think they may do not imply each other, especially when the underlying measure space $X$ is not complete).
Also, in the first theorem, $X$ is assumed to be a finite measure space, but in the second theorem, no such assumption is needed.
So here are my questions:
What is the precise relation between measurability and strong measurability?
Does the slogan "$f$ Bochner integrable iff $|f|$ integrable" holds for any measure space $X$? What technical conditions are needed?
Thank you!