TITLE: $u\in W^{1,p}(0,1)$ is equal a.e. to an absolutely continuous function?
QUESTION [11 upvotes]: 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.

REPLY [2 votes]: The answer posted by Jose27 is correct. For more details and related things, see the reference here: http://www.iadm.uni-stuttgart.de/LstAnaMPhy/Weidl/fa-ws04/Suslina_Sobolevraeume.pdf, especially Theorem 5.