I want to prove that:
If f is a continuous function, and ${X_n \to X} $ a.e then ${f(X_n) \to f(X)}$ a.e
I want to prove that:
If f is a continuous function, and ${X_n \to X} $ a.e then ${f(X_n) \to f(X)}$ a.e
As Stefan commented, the 'event' (set of $\omega$'s) that $X_n\to X$, is included in the 'event' when $f(X_n)\to f(X)$, just because of continuity.
So (using $\mu$ for the measure now), $0 = \mu(X_n \not\to X) \ge \mu(f(X_n)\not\to f(X)) \ge 0$