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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. – Tim Duff Sep 25 '12 at 11:44
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@Tim Duff : You would then have to prove equivalence with the definition using simple functions, won't you? – Doubting Thomas Sep 25 '12 at 11:49
That's directly computable from the definition you're using. – Tim Duff Sep 25 '12 at 11:56

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