Lebesgue point

Builds on Lebesgue measure Integrable function Open ball

Definition

Let $n \ge 1$ , let $\lambda$ be Lebesgue measure on $\R^n$ , and let $f \colon \R^n \to \R$ be locally integrable. For $x \in \R^n$ and $r > 0$ , let $B(x, r)$ be the open ball of centre $x$ and radius $r$ . A point $x \in \R^n$ is a Lebesgue point of $f$ if $\lim_{r \to 0^+} \frac{1}{\lambda(B(x, r))} \int_{B(x, r)} \abs{f(y) - f(x)} \, d\lambda(y) = 0.$

Remarks

Example: if $f$ is continuous at $x$ , then $x$ is a Lebesgue point. Given $\varepsilon > 0$ , choose $r_0$ with $\abs{f(y) - f(x)} < \varepsilon$ on $B(x, r_0)$ ; then every average with $r < r_0$ is at most $\varepsilon$ .

Non-example: let $n = 1$ and $f = \mathbf{1}_{[0, \infty)}$ . At $x = 0$ , the integrand equals $1$ on $(-r, 0)$ and $0$ on $[0, r)$ , so the average is $1/2$ for every $r$ , and $0$ is not a Lebesgue point.

The set of Lebesgue points depends on the chosen function, not only on its class modulo null sets. Changing $f$ at one point $x$ can destroy the Lebesgue point property at $x$ but at no other point.

Reading path

To understand this page from the ground up, read these 7 pages in order. The foundations come first.

Show the path as a graph

Each page uses the pages above it. The colour of the bar on a box shows how strong the evidence is, and the box also gives it as text. The list has the same pages.

no evidence level

Tag	`AWFXHJXGKE`
URI	`m2://AWFXHJXGKE`
Kind	definition
Writing	W0 stub
MSC	28A15
Defines	Lebesgue point
Uses	Lebesgue measure, Integrable function, Open ball
Changed	2 November 2026

Lebesgue point

Definition

Remarks

Reading path

Further reading

What this page needs for W1

What links here