What does it mean for a random variable to be bigger than another?

Question

I came across the following question while self-studying:

$$\text{Show that if } X \le Y \text{ then } E(X) \le E(Y)$$

Does it mean, for instance, that every value of $Y$ is bigger than every value of $X$?

Answer 1

If $X$ and $Y$ are random variables with the same underlying probability space $\langle\Omega,\mathcal A,\mathsf{P}\rangle$ then note that $X$ and $Y$ are measurable functions $\Omega\to\mathbb R$.

In this context $X\leq Y$ then stands for:$$\forall\omega\in\Omega\left[X(\omega)\leq Y(\omega)\right]$$

A consequence of that is:$$\mathsf{P}(X\leq Y)=1$$or in words: $X\leq Y$ almost surely.

Another consequence - requiring that for both the expectation exists - is:$$\mathsf EX\leq\mathsf EY$$as is mentioned in your question.