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Let $ (f_n)$ be a sequence of measurable functions on $ [0,1]$ such that $\int_{0}^1 |f_n|^2 dm \le C$.

Assume that there exist a function $f$ such that $\int_{0}^1 |f_n-f|dm \to 0$ as $n \to \infty $

show that $\int_{0}^1 |f|^2 dm \le C$ and it is true that $\int_{0}^1 |f_n-f|^2dm \to 0$ as $n \to \infty $ ?

I have no idea how to begin, any hints to start me off would be appreciated. Thanks

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The conclusion is false. Please check if your question is correctly formulated. –  23rd Dec 8 '12 at 16:32
    
@ richard, got the mistake, thanks –  jake Dec 8 '12 at 17:10
    
You are welcome! –  23rd Dec 8 '12 at 17:22

1 Answer 1

up vote 2 down vote accepted

Begin extracting a subsequence $\{f_{n_k}\}$ such that for all $k$, $$\int_{[0,1]}|f_{n_k}-f|dm< 4^{-k},$$ then that the sequence $\{f_{n_k}\}$ converges almost everywhere to $f$.

The first assertion is a consequence of Fatou's lemma.

The second one is not true: take $f_n:=\sqrt n\chi_{(0,n^{-1})}$, whose $L^2$ norm is $1$, and $f=0$. Then $\int_{(0,1)}f_n=n^{-1/2}\to 0$.

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