Consider a measurable function $f: X \to \mathbb{R}$. Let $A$ be a measurable set s.t. $\mu (A) > 0$. Assume that $$\int_A f \, \mathrm{d}\mu > 0.$$ Is there a measurable set $B \subset A$, $\mu (B) > 0$ such that $f(b)>0$ for all $b \in B$?
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Well, $B = \{x\in A:f(x) > 0\}$ is the maximal set of such kind. Want to show that $\mu(B) > 0$? Assume contrary.
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Ah, the elementary thing I was missing is: http://math.stackexchange.com/questions/293801/integral-over-null-set-is-zero – Matias Heikkilä Oct 27 '15 at 08:28