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I noted a definition of Lebesgue integral Here.

This might be masochism; but how would one prove additivity? This is an "exercise for the reader" in the book of Lieb and Loss, "Analysis".

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    If you already have the monotone convergence theorem then the proof is the same.2012-09-25
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    @Tim Duff : You would then have to prove equivalence with the definition using simple functions, won't you?2012-09-25
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    That's directly computable from the definition you're using.2012-09-25

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