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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?

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    By $\{f_{n}/n\in N\}$ do you mean $\{f_{n}:n\in\mathbb{N}\}$?2011-08-18
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    What does $\mu \sim \mu \circ F$ mean?2011-08-18
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    Yes, I mean the natural numbers, I couldn't right the natural numbers symbol for some reason.2011-08-18
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    μ∼μ∘F means that for every measurable set A, $\mu (A)=0$ iff $\mu (F(A))=0$2011-08-18
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    @Davide: since Arnold stated it as **iff**, I think the two are equivalent. But I am not sure that "iff" is in fact what he wants.2011-08-18

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