Assume we have $f\in L^1(K)$ for some compact set $K \subseteq \mathbb{R}$ and we have some sequence $(f_n)_{n \in \mathbb{N}} \in C^\infty(K) \cap L^1(K)$ where there exists $M>0$ such that $\lvert f_n(x) \rvert \leq M$ and $\lvert f(x) \rvert \leq M$ for every $n \in \mathbb{N}, x \in K$, such that $$ \lim_{n \rightarrow \infty} \int_{K} \lvert f_n(x) - f(x) \rvert~\mathrm{d}x = 0. $$ Now we introduce $g \in C^\infty\left(K_0\right) \cap L^1\left( K_0 \right)$ bounded where $K_0 := K \times [0, 1]^2$ such that $g\left( \mathbb{R}^3 \right) \subseteq K$. I am wondering whether even $$ \lim_{n \rightarrow \infty} \int_K \int_{[0, 1]^2} \left \lvert f_n(g(s, x)) - f(g(s, x)) \right \rvert~\mathrm{d}s~\mathrm{d}x = 0 $$ holds. To show the latter, I thought about passing to a subsequence of $f_n$. Then I get some subsequence such that $f_n \rightarrow f$ pointwise in $K \setminus N$ where $N$ has measure $0$. To now apply dominated convergence to the latter integral, I would need that $g(s, x) \in K \setminus N$ for almost every $(s, x) \in K \times [0, 1]^2$. But here I am stuck.
I would be happy to get any type of help or references of similar results. Thank you!