If E is measurable and f a nonnegative measurable function on E, then for every $0 \leq t \leq \int_E f < \infty$, there exists a measurable set $A \subseteq E$ with $\int_A f = t$.
I can't figure out the right way to start this problem. It seems like I could work from constant functions to simple functions and use some kind of limiting argument on those to get general nonnegative functions if I knew a similar property held for the Lebesgue measure (specifically, if $|E| = M$ then for every $0 \leq t \leq M$ there exists a subset A of E with $|A| = t$), and I think that such a property does hold, but it's not given and I can't figure out how to prove that, either.
Is there an easier way to do this than trying to work up from case of simple functions? Or, is there a way to handle the simple-function case without assuming the property holds for the Lebesgue measure?