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Rudin defined the integral of a nonnegative measurable function f as the supremum of a set of integrals of simple functions less than f.

With the assumption of f being bounded, how could I define the integral of f in terms of infimum of a set of integrals of simple functions greater than f. ?

What I got so far: Let f<=M, then integral of f is <= integral of M. M is a constant function hence is measurable and simple. Should I show the integral of M is the infimum of set?

Or should I start working on the integral of M-f, which is nonnegative measurable. The integral of M-f hence can be expressed as the sup of a set of integrals of simple functions less than M-f.

This definition appears in a paper which skips the proof. I would like a solid argument of it.

Please help!

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Considering $M-f$ seems like the thing to do. – Hagen von Eitzen Apr 7 '13 at 19:38
Thanks. But I am stuck on how to transfer from sup to inf if I use the sup definition for the integral of M-f. Could you please help me with the details? – user65214 Apr 7 '13 at 19:56

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