Lebesgue's density theorem

In mathematics, Lebesgue's density theorem states that for any Lebesgue measurable set $A\subseteq \mathbb {R} ^{n}$, the "density" of $A$ is 0 or 1 at almost every point in $\mathbb {R} ^{n}$. Additionally, the "density" of $A$ is 1 at almost every point of $A$. Intuitively, this means that the boundary of $A$, the set of points in $A$ for which all neighborhoods are partially in $A$ and partially outside $A$, is of measure zero.

Lebesgue's density theorem, applied to the inside of a square, its corners, edges, inside, and outside

Statement

Let $\mu$ be the Lebesgue measure on the Euclidean space and $A\subseteq \mathbb {R} ^{n}$ be a Lebesgue measurable set. Let $x\in \mathbb {R} ^{n}$ and let $B$_ε$(x)$ denote the open ball of radius $\varepsilon$ centered at $x$. Define the density at a point $x$

$$\qquad \qquad d_{A}(x)=\lim _{\varepsilon \to 0}{\frac {\mu (A\cap B_{\varepsilon }(x))}{\mu (B_{\varepsilon }(x))}}$$

Lebesgue's density theorem—For a measurable set $A\subseteq \mathbb {R} ^{n}$, the density of $A$ is 0 or 1 almost everywhere. If $0<\mu (A)<\infty$, then there are always points of $A\subseteq \mathbb {R} ^{n}$ That is, the set of points which do not have density 0 or 1, $\{x\in \mathbb {R} ^{n}\mid d_{a}(x)\notin \{0,1\}\}$, has measure 0. Furthermore if $0<\mu (A)<\infty$, then there are always points of $A$ where the density either does not exist or exists but is neither 0 nor 1.

For example, given a square in the plane, the density at every point inside the square is 1, on the edges is 1/2, and at the corners is 1/4. 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 of measure zero.

The Lebesgue density theorem is a particular case of the Lebesgue differentiation theorem.

Thus, this theorem is also true for every finite Borel measure on $A\subseteq \mathbb {R} ^{n}$ instead of Lebesgue measure, as proven in sections 2.8–2.9 of Federer's Geometric Measure Theory, 1969.

See also

• Lebesgue differentiation theorem – Mathematical theorem in real analysis

References

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