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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, chapter 11 –  Michael Greinecker Feb 19 '12 at 18:31
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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 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. –  paul garrett Jul 27 '12 at 23:54
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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.

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