# Let the nonnegative measurable function f be bounded. Prove that the integral of f is the infimum of a set.

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!

-

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