Lebesgues Differentiation Theorem

Let \(f\in L^1_{\text{loc}}(\mathbb{R}^n)\). Then for almost every \(x_0\in \mathbb{R}^n\) we have for \(r\to 0^+\)

\begin{equation*} \frac{1}{\lvert B_r(x_0)\rvert}\int_{B_r(x_0)} f \to f(x_0). \end{equation*}

A point which suffices the above property is called Lebesgue point of \(f\).