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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 $\mu$.

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    Aliprantis & Border, [Infinite Dimensional Analysis](http://www.amazon.com/Infinite-Dimensional-Analysis-Hitchhikers-Guide/dp/3540295860/ref=sr_1_1?ie=UTF8&qid=1329676247&sr=8-1), chapter 112012-02-19
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    For some on-line, free things about vector-valued integration, though not addressing "Bochner integrals", I have notes on my functional analysis page http://www.math.umn.edu/~garrett/m/fun/ The context for the vector-valued integration (in course notes near the bottom of the page) may be fancier than you need/want, but I'd claim that it's not really so fancy... and that the fact that a "weak/Gelfand-Pettis" integral is easily proven to exist waaay generally argues that the discussion must not be tooo complicated.2012-07-27

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