Inequality of expectation

Question

Is it true that if $E[X] \leq E[Y]$, then $X \leq Y$ a.e?

My idea of proof: Consider $A=\{Y < X\}$. $\int_{A}Y-X \leq 0$. But, by assumption, $E[X] \leq E[Y]$. Hence, $\mathbb{P}(A)=0$

Answer 1

Take $X$ with standard normal distribution and $Y=2X$ to get a counterexample.