Suppose that $X_n$ is a sequence of random variables such that $X_n \rightarrow X$ either almost surely or in $L^p$ or both.
Now consider the sequence of conditional random variables $f(X_n) | X_n \in B$ for some measurable set $B$ and some nice function $f$ (we can assume it's Lipschitz to make everything easy). Do I still preserve convergence properties for this conditional random variable, i.e. does $f(X_n) | X_n \in B \rightarrow f(X) | X \in B$ in some sense (a.s. or in $L^p)$?
I'm stuck verifying this by using the definition of almost sure convergence or $L^p$ convergence.
For example, regarding almost sure convergence, I have to check that $P(\lim_{n\rightarrow \infty} f(X_n)|X_n \in B = f(X)|X\in B)$ which looks strange from a notation perspective because of the conditional. Or should I resort to using conditional expectation?