# Lebesgue point

2010 Mathematics Subject Classification: Primary: 26B05 Secondary: 28A2049Q15 [MSN][ZBL]

Let $f: \mathbb R^n \to \mathbb R^k$ be an absolutely locally integrable function (with respect to the Lebesgue measure $\lambda$). A Lebesgue point $x$ for $f$ is a point where $\lim_{r\downarrow 0} \frac{1}{\lambda (B_r (x))} \int_{B_r (x)} |f(y)-f(x)|\, dy = 0\, .$ Note that a Lebesgue point is, therefore, a point where $f$ is approximately continuous. Viceversa, if $f$ is essentially bounded, then any point of approximate continuity is a Lebesgue point.

The following theorem of Lebesgue holds (see Section 1.7.2 of [EG]).

Theorem 1 Let $f$ be as above. Then $\lambda$-a.e. $x$ is a Lebesgue point for $f$.

The set of Lebesgue points of $f$ is called Lebesgue set.