Assume you have the following setting:
Let $\mathbb{P}$ a probability measure and a sequence of real-valued random variables with convergence $X_n\stackrel{L^1(\mathbb{P})}{\rightarrow} X$. Assume $f\in C(\mathbb{R})$. Does the following convergence holds:
$f(X_n)\stackrel{L^1(\mathbb{P})}{\rightarrow} f(X)$
If not is it enough to assume that $X_N, X$ or $f$ are bounded? I guess then it is just dominated convergence.