Would anyone be able to check if I've missed anything for my answer to the following question. The requirements I've noted so far is that:
For every bounded and continuous $f: \mathbb{R} \rightarrow \mathbb{R}$ , $F_{X_n}(x) \rightarrow F_X(x)$ at every $x$ where the limit distribution function $F_X$ is continuous. As a consequence of it converging in $L^1$ , it must also converge in measure by the following inequality
$\mu\left(\left\{x \in X:\left|f_n(x)-f(x)\right|>\varepsilon\right\}\right) \leq \frac{1}{\varepsilon} \int\left|f_n-f\right| d \mu$.