More general Continuous Mapping Theorem I wonder whether there might be a more general version of the Continuous Mapping Theorem (CMT), at least under certain conditions. What we know from the basic CMT is that 
$$ X_n \rightarrow X \text{ and } g \text{ continuous} \Rightarrow g(X_n) \rightarrow g(X),$$
where convergence can mean convergence of a real sequence, convergence in distribution, in probability and so on. 
Now what if instead of a single function $g$ I have a sequence $(g_n)$ of continuous functions that converges (probably uniformly) to a continuous function $g$. Are there any conditions so that I can also conclude that 
$$ g_n(X_n) \rightarrow g(X)?$$
 A: Have a look at Theorem 1.11.1 in Van Der Vaart and Wellner, https://link.springer.com/book/10.1007/978-1-4757-2545-2: Copied here:
1.11.1 Theorem (Extended continuous mapping). Let $\mathbb{D}_{n} \subset \mathbb{D}$ and $g_{n}: \mathbb{D}_{n} \mapsto \mathbb{E}$ satisfy the following statements: if $x_{n} \rightarrow x$ with $x_{n} \in \mathbb{D}_{n}$ for every $n$ and $x \in \mathbb{D}_{0}$, then $g_{n}\left(x_{n}\right) \rightarrow g(x)$, where $\mathbb{D}_{0} \subset \mathbb{D}$ and $g: \mathbb{D}_{0} \mapsto \mathbb{E}$. Let $X_{n}$ be maps with values in $\mathbb{D}_{n}$, let $X$ be Borel measurable and separable, and take values in $\mathbb{D}_{0}$. Then
(i) $X_{n} \leadsto X$ implies that $g_{n}\left(X_{n}\right) \leadsto g(X)$;
(ii) $X_{n} \stackrel{\text { P* }}{\rightarrow} X$ implies that $g_{n}\left(X_{n}\right) \stackrel{\text { P* }}{\rightarrow} g(X)$;
(iii) $X_{n} \stackrel{\text { as: }}{\rightarrow} X$ implies that $g_{n}\left(X_{n}\right) \stackrel{\text { as* }}{\rightarrow} g(X)$.
