Originally Posted by

**iknowone** Let $f$ be a nonnegative Riemann integrable function on [a,b].

Prove that $int_{a}^{b}f(x) dx = 0$

if and only if

$E = {x \in [a,b] | f(x) > 0}$ is a null set.

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<= If E is a null set I want to be able to define a partition

that allows me to control the contribution to the integral on those

intervals which contain points in E by arguing that these intervals have

arbitrarily small length and f(x) is bounded. Then I can control the

contribution to the integral on those intervals which do not contain

points in E by observing that the function is equal to 0 at these points.

But given that E is a null set I get a cover of E with arbitrarily small

length. But this cover is the countable union of intervals. I need to refine

this to a finite subcover if I hope to construct a partition as described

above. E needn't be compact as far as I can tell thus a finite subcover is

not guaranteed.

=> No idea.