Positive subset of sigma-algebra

Question

I let G be a sub-σ-algebra of F. Suppose that an integrable G-measurable random variable Y satisfies that,

$$ \int_A YdP\geq 0, \qquad \forall A\in G $$

Now I want to show that $$Y\geq 0$$ a.s.

This makes sense intuitively for me, but not mathematically

Answer 1

Let $A_n=\{Y\leq -\frac{1}{n}\}$, which is $\mathcal G-$measurable. Then $$0\leq \int_{A_n}Y\leq -\frac{1}{n}\mathbb P(A_n),$$ and thus $\mathbb P(A_n)=0$ for all $n$. Therefore, $$0=\lim_{n\to \infty }\mathbb P(A_n)=\mathbb P\left(\bigcup_{n\geq 1}A_n\right)=\mathbb P\{Y<0\}.$$ Therefore $Y\geq 0$ a.s.