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.