Lebesgue density theorem

Let μ be the Lebesgue measureMathworldPlanetmath on n, and for a measurable setMathworldPlanetmath An define the density of A in ϵ-neighborhoodMathworldPlanetmath of xn by


where Bϵ(x) denotes the ball of radius ϵ centered at x.

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


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 μ(A)>0 and μ(nA)>0, then there are always points of n where the density is neither 0 nor 1 [1, Lemma 4].


  • 1 Hallard T. Croft. Three lattice-point problems of Steinhaus. Quart. J. Math. Oxford (2), 33:71–83, 1982. http://www.emis.de/cgi-bin/zmen/ZMATH/en/quick.html?type=html&an=0499.10035Zbl 0499.10035.
Title Lebesgue density theorem
Canonical name LebesgueDensityTheorem
Date of creation 2013-03-22 13:21:02
Last modified on 2013-03-22 13:21:02
Owner bbukh (348)
Last modified by bbukh (348)
Numerical id 7
Author bbukh (348)
Entry type Theorem
Classification msc 28A75