Is it possible that a random variable is greater 0 a.e. and its expectation is still 0?

Question

I am trying to prove a simple question in probability. It is quite obviously, but somehow I could not prove it.

Suppose that a random variable $X > 0$ a.e.. It is possible that ${E}(X)=0$?

Thank you in advance.

Answer 1

Hint. Let $E_n := \{X \ge \frac 1n\}$. The $E_n$ are measurable and $\{X > 0\} = \bigcup_n E_n$. Now argue as follows:

(i) Show that we cannot have $\mathbf{P}(E_n) = 0$ for all $n$ using that $\mathbf{P}(X > 0) = 1$.

(ii) Use that $X$ is positive a. e. and hence $\mathbf E(X) \ge \frac 1n \mathbf{P}(E_n)$ for all $n$.