# An alternative definition for integral of a nonnegative measurable function in terms of infimum

How could I show "integral of a nonnegative measurable function f could be defined as the infimum of a set of integrals of simple functions g with f<=g for all g".

We could assume f is bounded by M. Then M-f is nonnegative measurable. How to proceed further from here?

Help is appreciated!

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Take a look at $f(x) = e^{-x}$ on $[0,\infty)$. Do you see the problem?

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If the function is bounded above on a bounded interval, your formulation would work. – ncmathsadist Apr 7 '13 at 2:43
If I assume f is bounded above on any measurable set A with the measure of A being finite, would my formulation be correct? If so, any hint on how to prove it? – user65214 Apr 7 '13 at 2:58