Let $\Omega\subset \mathbb{R}^n$ an open set and $u:\Omega\to \mathbb{R}$ be a (locally) $L^1$-function. Then it is well known that the Lebesgue differentiation theorem holds: For almost every $x\in \Omega$, $$\frac{1}{|B_r(x)|} \int_{B_r(x)} (u(y) - u(x)) d y \to 0$$ as $r\to 0$. My question is if it is true that $$\frac{1}{|B_r(x)|} \int_{B_r(x)} |u(y) - u(x)| d y \to 0$$ as $r\to 0$ for almost every $x\in \Omega$. I am doubtful about this, but the universal counterexample in measure theory - the (characteristic function of the) fat Cantor set - doesn't work here. If it is actually true, then it could probably further be generalized to $$\frac{1}{|B_r(x)|} \int_{B_r(x)} |u(y) - u(x)|^p d y \to 0$$ if $u\in L^p(\Omega)$ and $p\ge 1$. Actually that is what I'm interested in.
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4$\begingroup$ Such a point is called a "Lebesgue point" of $u$. Look for that in your textbook. Here it is in Wikipedia: en.wikipedia.org/wiki/Lebesgue_point $\endgroup$– Gerald EdgarCommented Oct 19, 2011 at 14:12
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$\begingroup$ To sketch the standard argument: Since countable intersections of full-measure sets is a full-measure set, a countable set of integrable functions admits a common full-measure set $S\subset\Omega$ of differentiation (that is, such that the first formula you wrote holds for every function of the family and for every $x\in S$). Consider in particular the functions $u_q(y):=|u(y)-q|$, $q$ a rational number. Then it is easy to see that for all $x\in S$ the second formula you wrote is true. Check for instance Wheeden- Zygmund, Measure and integral. $\endgroup$– Pietro MajerCommented Oct 19, 2011 at 17:05
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It's true that $$\frac{1}{|B_r(x)|}\int_{B_r(x)} |u(y)-u(x)|^p\to 0$$ for $r\to 0$. For $p=1$ this is standard (Somehow I always overlooked the absolute value signs in the definition of Lebesgue points), and the same proof carries over for $p>1$ (at least the proof used in Ziemer, "Weakly differentiable functions", where variants of this theorem for Sobolev spaces are discussed as well).