Lebesgue's density theorem

Let μ be the Lebesgue measure on the Euclidean space Rn and A be a Lebesgue measurable subset of Rn.

Define the approximate density of A in a ε-neighborhood of a point x in Rn as where Bε denotes the closed ball of radius ε centered at x. Lebesgue's density theorem asserts that for almost every point x of Rn the density exists and is equal to 0 or 1.

In other words, for every measurable set A, the density of A is 0 or 1 almost everywhere in Rn.

The set of points in the plane at which the density is neither 0 nor 1 is non-empty (the square boundary), but it is negligible.

This article incorporates material from Lebesgue density theorem on PlanetMath, which is licensed under the Creative Commons Attribution/Share-Alike License.