Let $1 \leq p < \infty$ and $f \in L^p_{\mathrm{loc}}(\mathbb{R}^n)$. Then for $\mathcal{L}^n$-almost every $x \in \mathbb{R}^n$, \begin{align*} \lim_{r \to 0} \frac{1}{\mathcal{L}^n(B(x,r))} \int_{B(x,r)} |f(y) - f(x)|^p \, d\mathcal{L}^n(y) = 0. \end{align*} In particular, every $x$ in the Lebesgue set of $f$ is also a Lebesgue point in this $L^p$ sense.