How to prove $Z\geq 0, \text{ a.s.}$ when $\int_A Z\,dP\geq0$ for any $A$?

Question

How to prove $Z\geq 0, \text{ a.s.}$ when $\int_A Z\,dP\geq0$ for any $A$?

The $P$ is probability.

Answer 1

Take $A$ to be $\{\omega : Z(\omega) < 0\}$. The assumptions imply $\int_A Z \,dP \geq 0$, so the nonpositive function $Z 1_A$ has integral zero. This implies $Z 1_A = 0$ a.s. Since $Z 1_A < 0$ on $A$, it follows that $A$ must have measure zero.