# How badly can the Lebesgue differentiation theorem fail?

Suppose $$f:\mathbb{R}^n\to\mathbb{R}$$ is integrable. Is it true that $$\lim_{r\to 0}\frac{\displaystyle\int_{B_r(0)}f(y)~\mathrm dy}{r^{n-1}}=0 \quad ?$$ This is obvious if $$0$$ is a Lebesgue point of $$f$$ or if $$n=1$$, but I would like to know if it's true in general.

• $|x|^{-\alpha}$ with $1 \leq \alpha <n$ is an example when $n>1$. Sep 5, 2022 at 11:10
• Thanks, it was easier than I thought. You should post your comment as an answer so I can accept it. Sep 5, 2022 at 11:16

Metafune has given an example of the limit failing to be $$0$$ at a particular point - namely for $$n > 1$$, the function $$|x|^{-\alpha}$$, with $$1 \leq \alpha < n$$ has that limit equal to $$\infty$$ at $$0$$.
In general the limit in question is zero $$\mathcal H^{n-1}$$-a.e, where $$\mathcal H^{n-1}$$ denotes the $$n-1$$ dimensional Hausdorff measure.