Let $G$ be a sigma algebra. $Y$ is $G$ measurable and E|Y| < $\infty$ . We also have that $E[ZY] \ge 0$ for all $Z$ such that $Z$ is a bounded, $G$ measurable random variable. Prove that $P(Y \ge 0)=1$.
In my attempt, I think I should show P (Y $< 0$) = $0$ . I have written {$Y < 0$} is the intersection of {$Y<1/n$} for n =1 to infinity. So when P($Y<0$) >0 there is at least one $n_0$ that makes P{$Y<1/n_0$} .
Also by Markov , $E[Y;Y<1/n]\le(1/n)P(Y<1/n)$
I also tried to write $Y = Y_+-Y_-$ and want to manipulate when Y <0
But it seems I have ran out of idea. So please help.
What would you get, if you tried $Z = I _{\{Y<0\}}$? Then you might use the property, that if $\alpha \leq 0$ and $E \alpha \geq 0$, then $\alpha = 0$ almost surely.