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Let $f,f_1,f_2,\ldots\colon\mathbb{R}\rightarrow\mathbb{R}$ be continuous functions in $L^2(\mathbb{R})$. Suppose that $\|f_n-f\|_2\rightarrow 0$ as $n\rightarrow 0$. Is it true that $f_n(x)\rightarrow f(x)$ for almost every $x$?

Without the continuity assumption, I remember this is not true. But what about with continuity?


marked as duplicate by Daniel Fischer, Dan Rust, Namaste, Did, Nate Eldredge Dec 5 '13 at 21:30

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  • $\begingroup$ Not necessarily. $\endgroup$ – Daniel Fischer Dec 5 '13 at 20:49

No, for all $k\in \mathbb N$ and $j=0,\dots,k-1$ you can find a function $f_{k,j}$ such that $f_{k,j}$ is continuous, $f_{k,j}(x) \ge 1$ for $x \in [j/k,(j+1)/k]$ and $\int f^2_{k,j}\le 2/k$. The sequence $f_{1,0},f_{2,0},f_{2,1},f_{3,0},f_{3,1}\dots$ has the desired properties.


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