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.



How badly can the Lebesgue differentiation theorem fail?
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 $0$.


However, you can still get some kind of affirmative result.


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.


This is Theorem 2.10 in *Measure Theory and Fine Properties of Functions* by Evans and Gariepy (2015 version).


