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My question is the following. If a sequence of absolutely continuos functions $\{f_n\}$ converge to zero in $L^p(S)$ ($S$ has finite measure), does it follow that the $f_n \to 0$ everwhere ?

Siminore
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yasin
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    Do you know an example when the $f_n$ are not absolutely continuous? You should be able to modify it by replacing the jumps with sufficiently steep line segments. – Nate Eldredge Sep 03 '14 at 14:06
  • Certainly not everywhere. Consider e.g. $$f_n:\left[0,1\right]\to\mathbb{R},x\mapsto\begin{cases} n\left(\frac{1}{n}-x\right), & x\in\left[0,\frac{1}{n}\right),\ 0, & x\geq\frac{1}{n}. \end{cases}$$ Note that all $f_n$ are Lipschitz, hence absolutely continuous, but $f_n(0) = 1$ for all $n$. Even almost everywhere is not true, as indicated by @NateEldredge. – PhoemueX Sep 03 '14 at 15:14

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