## You are here

HomeLebesgue density theorem

## Primary tabs

# Lebesgue density theorem

Let $\mu$ be the Lebesgue measure on $\mathbb{R}^{n}$, and for a measurable set $A\subset\mathbb{R}^{n}$ define the density of $A$ in $\epsilon$-neighborhood of $x\in\mathbb{R}^{n}$ by

$d_{\epsilon}(x)=\frac{\mu(A\cap B_{\epsilon}(x))}{\mu(B_{\epsilon}(x))}$ |

where $B_{\epsilon}(x)$ denotes the ball of radius $\epsilon$ centered at $x$.

The Lebesgue density theorem asserts that for almost every point of $A$ the density

$d(x)=\lim_{{\epsilon\to 0}}d_{{\epsilon}}(x)$ |

exists and is equal to $1$.

In other words, for every measurable set $A$ the density of $A$ is $0$ or $1$ almost everywhere. However, it is a curious fact that if $\mu(A)>0$ and $\mu(\mathbb{R}^{n}\setminus A)>0$, then there are always points of $\mathbb{R}^{n}$ where the density is neither $0$ nor $1$ [1, Lemma 4].

# References

- 1 Hallard T. Croft. Three lattice-point problems of Steinhaus. Quart. J. Math. Oxford (2), 33:71–83, 1982. Zbl 0499.10035.

Type of Math Object:

Theorem

Major Section:

Reference

## Mathematics Subject Classification

28A75*no label found*

- Forums
- Planetary Bugs
- HS/Secondary
- University/Tertiary
- Graduate/Advanced
- Industry/Practice
- Research Topics
- LaTeX help
- Math Comptetitions
- Math History
- Math Humor
- PlanetMath Comments
- PlanetMath System Updates and News
- PlanetMath help
- PlanetMath.ORG
- Strategic Communications Development
- The Math Pub
- Testing messages (ignore)

- Other useful stuff
- Corrections