# 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 $$g_n(X_n) \rightarrow g(X)?$$

• The standard approach is to look at the following identities and check what can be deduced from them: $$g_n(X_n) - g(X) = g_n(X_n) - g_n(X) + g_n(X) - g(X)$$ and $$g_n(X_n) - g(X) = g_n(X_n) - g(X_n) + g(X_n) - g(X)$$ -- the triangle inequality is your friend, here. Commented Apr 20, 2020 at 18:24

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 $$g_{n}\left(X_{n}\right) \stackrel{\text { as* }}{\rightarrow} g(X)$$.