We have proven the following in class:
If $X$ is a finite random variable with $X\geq 0$ then $$E(X)=0 \iff P(X=0)=1$$ (By finite I meant that the range has finitely many elements).
Does it hold for any non-negative random variable $X:\Omega\to\mathbb R_{\geq0}$?
(I've tried proving it with the indicator function yet it didn't help.)
Best Answer
Note that, for every $x\gt0$, $$ X\geqslant x\mathbf 1_{X\geqslant x}. $$ (This is the step where one uses that $X\geqslant0$ almost surely.) It follows that $$ E(X)\geqslant xP(X\geqslant x). $$ If $E(X)=0$ this inequality implies that $P(X\geqslant x)=0$. Finally, $$ [X\gt0]=\bigcup_{n\geqslant1}[X\geqslant1/n], $$ hence, if every event in the RHS has probability zero the event on the LHS has probability zero.