Reference request: Continuous Mapping Theorem

Question

I'm looking for a reference (preferably, a textbook) for the Continuous Mapping Theorem, as stated in the Wikipedia page. In particular, I'm interested in point 3., concerning a.s. convergence.

I've looked into the reference suggested in the Wikipedia page, namely P. Billingsley, "Convergence of Probability Measures" (1999), but it seems to me that the statement (Theorem 2.7, p. 27) concerns weak convergence only.

Any help is very welcome! Thanks!

Answer 1

Almost sure convergence is trivial.

Note that $$\{\omega\mid X_n(\omega) \to X(\omega)\} \subset \{\omega \mid g(X_n(\omega)) \to g(X(\omega))\}$$

where $X_n,X$ are random variables and $g$ is continuous. Then, take probabilities of both sides of the inequality.