A random variable is almost surely equal to infinity

Question

Is it true that if $X > 0$ and $\mathbb{E}\left[e^{-X}\right] = 0$, then $\Pr(X = \infty) = 1$ a.s.? I've tried looking for a proof, but have not been able to find one yet.

Answer 1

If $\mathbb{P}(X \le k) = p_k$ for given finite $k$ then $\mathbb{E}\left[e^{-X}\right]\ge p_ke^{-k}$

Since $e^{-k}$ is positive for all finite $k$, this expectation cannot be $0$ unless $p_k=0$, for all finite $k$