Convergence in probability of random variables

Question

I want to prove that if $\xi_n \xrightarrow {\mathbb{P}} \xi$ and for some Borel function $\mathit{f}$, where $\mathbb{P}\{\omega \in \Omega|\ \mathit{f} \text{ is discontinuous at } \xi(\omega)\} = 0$ then $$\mathit{f}(\xi_n)\xrightarrow {\mathbb{P}} \mathit{f}(\xi)$$ I don't know where to start. I would be very grateful for help!

Answer 1

First, note that $X_n\xrightarrow{p}X$ iff every subsequence of $\{X_n\}$ has a sub-subsequence converging to $X$ a.s.

Take a subsequence $\{\xi_{n_k}\}$. We know that $\xi_{n_k}\xrightarrow{p}\xi$ and, thus, there is a subsequence $\{\xi_{n_{k_j}}\}$ converging a.s. to $\xi$. Since $\{f \text{ is discontinuous at }\xi\}$ is a null set, $f(\xi_{n_{k_j}})\to f(\xi)$ a.s.