I have the following question where I am required to show that a random variable is almost surely infinite:
Let $Y$ be a non-negative random variable, given that $$ E[e^{-Y}]=0, $$ show that $Y$ is almost surely infinite.
I know that "$Y$ is almost surely infinite" is equivalent to showing that $$ P[Y=\infty] = 1. $$
I have attempted to use the Markov Inequality $$ P[e^{-Y}\geq\epsilon] \leq \frac{E[|e^{-Y}|^r]}{\epsilon^r}, $$ we can pick $r=1$ and we know $e^{-Y}>0$ giving $$ P[e^{-Y}\geq\epsilon] \leq \frac{E[e^{-Y}]}{\epsilon} = 0 \implies P[e^{-Y}\geq\epsilon] = 0. $$ I think that the next step involves showing that $$ P[Y>k] = 1, $$ however I am unsure how to do so rigorously, and where I would go from here.
I would really appreciate any help steering me in the right direction. Thanks