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I know the standard way to define the integral of a nonnegative measurable function f by using the supremum. How would you do it by taking the infimum?

Here is the question:

Let the nonnegative measurable function f be bounded. Prove that the integral of f under any measurable set A (assume A has finite measure) is the infimum of the set of integrals of simple functions t with f<=t.

Any help would be appreciated!

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Assume $f(x)\le M$ for all $x$ and consider $M-f(x)$.

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  • $\begingroup$ Thx. Is it legitimate for me to say the integral of M-f(x) is the supremum of the set of integrals of simple functions? $\endgroup$ – user65214 Apr 6 '13 at 20:40

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