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.