Let $F\colon (0,1)\rightarrow (0,1)$ be a non-singular function with respect to the Lebesgue measure $\mu$ (so $\mu\sim\mu \circ F$). Let $\{ f_n : n \in \mathbb{N} \} \subset L^{2}([0,1])$ be a sequence of simple integrable functions and $f\in L^{2}([0,1])$ such that $f_n\to f $ in the $2$-norm. Is it correct that also $f_n\circ F\to f\circ F$ ?
If not, what are the conditions on $F$ such that this implication is correct?