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 $\mu \{x \in X : f(x)<0\}=0$ ?
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 $\mu \{x \in X : f(x)<0\}=0$ ?
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