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I studied the Lebesgue integral from Stein's book on Real Analysis.
In the book, he first considered the Lebesgue measure on $R^d$, defined measurable functions and then showed that they can be approximated by simple functions, the proof of this fact relied on the fact that $R^d$ is $\sigma $- finite, then he defined the integral for simple function and worked his way up from there, until he finally defined the Lebesgue integral on all non-negative measurable functions. Later, he introduced the idea of an abstract measure space then sketched how to generalize the integral he defined on $R^d$ to abstract measure spaces which are $\sigma$*-finite(In particular, he uses this fact to prove an analogue of the approximation theorem.) $$$$ **Is there an alternative approach which allows us to define the integral on general measure spaces?(Or at least another version of the approximation theorem which don't rely on $\sigma$*-finite-ness?)
If there is, can anybody sketch the idea or the main difference between the two approaches?(Or suggest a good quick read to get some insight into the matter.) **

Thanks in advance!

Burrrrb
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    In an arbitrary measure space you can define the integral of simple functions as usual. Then you consider only nonnegative (nn) simple functions. One can show quite easily that each nn measurable function $f$ can be pointwise approximated from below by a sequence of nn simple functions. Then you show that the sequence of their integrals is non-decreasing and define the integral of $f$ by the supremum. – amsmath Oct 16 '17 at 02:38

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DH Fremlin in his Measure Theory Volume II (2010) discusses integration in non-$\sigma$-finite spaces. He proposes an integral for semi-finite spaces. Integrals for spaces that are not at least semi-finite must contend with ill-behaved infinite sets. He proposes to transform such "bad" spaces to better behaved semi-finite, complete, and saturated spaces that rid themselves of the "bad" infinite sets while preserving the spaces's "good" properties. Fremlin's books on measure theory are free on the internet but are becoming somewhat hard to find.

Joseph
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