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I have a simple question on Sobolev space theory. Let $1\le p \le \infty$. How can one prove that every $u\in W^{1,p}(0,1)$ is equal a.e. to an absolutely continuous function and that $u'$ exists a.e. and belongs to $L^p(0,1)$?

Thank you for your assistance.

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    What does equal s.e. mean? Also don't you mean $W^{1,p}(0,1)$? If I'm guessing right, it looks like a basic theorem in Sobolev space theory...2012-12-02
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    I mean the function can be represented by a function that is a.e. equal to an absolutely continuous function.2012-12-09

2 Answers 2