Let $X$ be some space (eg. a manifold). Suppose that $f_n:X \to \mathbb{R}$ converges pointwise a.e. to $f:X \to \mathbb{R}$. Let $g_n:\mathbb{R} \to \mathbb{R}$ be continuous with $g_n \to g$ uniformly (in fact in $C^\infty$).
Is it true that $$g_n(f_n) \to g(f)$$ pointwise a.e?
I don't know how to do these double limit problems.
Since $g_n$ converges uniformly to $g$, we know that given $\epsilon > 0$, $\exists N$ so that $n \geq N \Rightarrow |g_n(x) - g(x)|<\epsilon$, $\forall x \in \mathbb{R}$. Therefore, given $x_0 \in X$, $\epsilon > 0$, since $f(x_0), f_n(x_0) \in \mathbb{R}$, $\exists N_1:n \geq N_1 \Rightarrow |g_n(f(x_0))-g(f(x_0))|<\epsilon/2$.
On the other hand, since $f_n$ converges pointwise to $f$, then given $\epsilon > 0$, \exists $N$ so that $n\geq N \Rightarrow |f_n(x) - f(x)|<\epsilon$. Therefore, for the same $x_0$ and $\epsilon$, $\exists N_2:n\geq N_2 \Rightarrow |f_n(x_0) - f(x_0) |< \epsilon / 2$.
If $N=\max \{N_1, N_2\}$, then given the very same $\epsilon$ and $x_0$, taking into account that $g$ is $C^{\infty}$ and therefore continuous, we've got that:
$$n \geq N \Rightarrow | g_n(f_n(x_0)) - g(f(x_0))| \leq |g_n(f_n(x_0)) - g(f_n(x_0))| + |g(f_n(x_0)) - g(f(x_0))| \leq \epsilon / 2 + \epsilon / 2 = \epsilon$$
So $g_n(f_n)$ converges pointwise to $g(f)$.