Q:

Is there a non-null subset where $f$ has the sign of its integral?

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 $b \in B$?

A:

Well, $B = \{x\in A:f(x) > 0\}$ is the maximal set of such kind. Want to show that $\mu(B) > 0$? Assume contrary.

