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Why is Lebesgue integration taught using positive and negative parts of functions?

Hey,

I am currently referring 'probability with martingales'. To develop lesbegue integration they have first defined it over simple functions.

where a simple function is defined as sigma { (a_i) * I_Ai }

where each a_i is positive, and Ai form a partition over the sample space. I_Ai is the indicator function taking value for elements in A_i and 0 else.

My question being why is the condition of positivity on a_i needed? To the best of my knowledge it isn't used in derivation f its properties.

Further then integral is defined separately for positive functions and then for general functions (writing it as difference of two positive functions)

My question is, why such different approach to it? Why not define it directly for general functions?

Thanks!

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  • $\begingroup$ I am not sure what kind of answer you expect. For convenience? $\endgroup$
    – Marc Palm
    Commented Mar 15, 2012 at 18:40
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    $\begingroup$ It's a long time since I studied this, but I think one initially wants to allow for integrals of positive functions to possibly take the value "plus infinity", as it makes various statements and proofs using monotonicity easier to handle. This is only possible if you rule out infinity minus infinity types of nonsense, hence one restricts to positive integrands. I seem to recall some related discussion of this on MO. $\endgroup$
    – Yemon Choi
    Commented Mar 15, 2012 at 18:45
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    $\begingroup$ QUite close to this question: mathoverflow.net/questions/25161 $\endgroup$
    – Gerald Edgar
    Commented Mar 15, 2012 at 18:59
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    $\begingroup$ I also feel I should stick up for P with M, as it was one of the texts for the course where I actually learned my measure theory. My suspicion, as seems to be borne out in the answers to which Gerald Edgar has linked, is that both ways of setting up integrals have their advantages and disadvantages. However, since I am not a probabilist, this is just conjecture on my part. $\endgroup$
    – Yemon Choi
    Commented Mar 16, 2012 at 0:57

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The positivity is absolutely not needed. Serge Lang's Real Analysis directly develops Lebesgue integration of Banach-space-valued functions, so his simple functions are not positive. Not only is it much cleaner than the $f_+-f_-$ business, it is also more general since the "standard" approach can only be extended to functions taking values in finite-dimensional vector spaces.

EDIT: I didn't see that Gerald's comment already links to that answer, sorry for the duplicate.

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