Reference request on vector-valued integration I'm looking for some nice, neat text which discusses the Bochner and Pettis approaches to integration of vector-valued functions. I'm not interested in the most general case, so the less technical the text the better. To be precise, the level of generality I'm interested in is integration of functions defined on some measure space $(X,\mathcal{M},\mu)$ taking values in some Banach space $V$, w.r.t. the measure $\mu$.

Well, it is not exactly what you ask for, but for the case $X=I$, an interval, there is a nice elementary introduction in 
Arendt, Batty, Hieber, Neubrander: Vector-valued Laplace Transforms and Cauchy Problems
It is not technical, and they give you the broad idea with references.
