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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 ...

Lebesgue's differentiation theorem states that if $x$ is a point in $\mathbb{R}^n$ and $f:\mathbb{R}^n\rightarrow\mathbb{R}$ is a Lebesgue integrable function, then the limit of $\frac{\int_B f d\...