If a sequence converges in L2 and we compose every function with a non-singular function, does it still converge?

Let $F:(0,1)\rightarrow (0,1)$ be a non-singular function with respect to the lebesgue measure $\mu$ (so $\mu\sim\mu \circ F$ ) . let $\lbrace f_n/n\in N\rbrace\subset L^{2}([0,1])$ be a sequence of simple integrable functions and $f\in L^{2}([0,1])$ such that $f_n\rightarrow f$ in the 2-norm. is it correct that also $f_n\circ F\rightarrow f\circ F$ ?

If not, what are the conditions on $F$ such that this inplication is correct?

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This question is at the homework level in a basic real analysis course. Did you try math.stackexchange.com MO is devoted to research level questions. – Bill Johnson Aug 18 2011 at 14:36