Does uniform convergence of $X_n$ imply $L^p$ convergence of $f(X_n)$?

Question

Suppose I have $X_n \sim F_n$, $X \sim F$ such that $\Vert F_n - F \Vert_\infty \rightarrow 0$, does this imply that $f(X_n) \xrightarrow{L^p} f(X)$ for any continuous $f$?

I know that, since we are dealing with probability measures/distribution functions, $X_n \xrightarrow{L^p} X$ holds.

I know that, for example, similar results hold for convergence in probability or weak convergence (using unif convergence implies $\xrightarrow{p}$, and then using CMT), but I was wondering if I can say something about $L^p$ convergence.

Answer 1

No, you can't say anything about that convergence.
The reasoning behind that is the different nature of these two types of convergence.
While the condition that you give provides a lot of info on the convergence in law, it gives nothing on the spatial relation between $(X_n;n \ge 0)$.
A good thought counterexample is when $(X_n; n \ge 0)$ are independent and identically distributed.