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**azdang** Given $\displaystyle (\Omega, \mathcal{A}, P)$, suppose X is a random variable with $\displaystyle X\geq0$ and E{X}=1. Definite $\displaystyle Q:\mathcal{A}->R$ by Q(A)=$\displaystyle E{(X1_A)}$. Show that if P(A) = 0, then Q(A)=0. Give an example that shows that Q(A)=0 does not in general imply P(A)=0.

I'm pretty sure that $\displaystyle E{(1_A)}=P(A)$, so I was thinking that maybe we could break Q(A) up into $\displaystyle E{(X)}E{(1_A)}$. However, this would be assuming that X and 1_A are independent random variables? Would this be true? If so, then it's very obvious why, if P(A)=0, then Q(A)=0. Anyone have any hints? Thanks!