Expectation of random variable with domain $\mathbb{N} \cup \{ \infty \}$

Question

Let us consider a random variable $X \colon \Omega \rightarrow \mathbb{N} \ \cup \{ \infty \} $. Is there a reasonable way to define the expectation value $E(X)$? How do I deal with the fact that $X$ could be equal to $\infty$? Is the expectation value always $\infty$ if $P(X=\infty)>0$? Thanks!