$(X,mu , mathcal F)$ a measure space , $f:Xtomathbb R$ measurable ; $int_A f dmu ge 0 , forall A in mathcal F$ , then $f ge 0$ a.e.?

Let $(X,\mu , \mathcal F)$ be a measure space and $f:X\to\mathbb R$ be a measurable function such that $\int_A f d\mu \ge 0 , \forall A \in \mathcal F$ , then is it true that $\mu \{x \in X : f(x)<0\}=0$ ?