Lebesgue's density theorem

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

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

${\displaystyle \qquad \qquad d_{\varepsilon }(x)={\frac {\mu (A\cap B_{\varepsilon }(x))}{\mu (B_{\varepsilon }(x))}}}$

Lebesgue's density theorem asserts that for almost every point ${\displaystyle x}$ of ${\displaystyle A\subseteq \mathbb {R} ^{n}}$ the density

${\displaystyle \qquad \qquad \qquad d(x)=\lim _{\varepsilon \to 0}d_{\varepsilon }(x)}$

exists and is equal to 0 or 1.

For every measurable set ${\displaystyle A}$, the density of ${\displaystyle A}$ is 0 or 1 almost everywhere¹. If ${\displaystyle 0<\mu (A)<\infty }$, then there are always points of ${\displaystyle A\subseteq \mathbb {R} ^{n}}$ where the density either does not exist or exists but is neither 0 nor 1.².

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 ${\displaystyle A\subseteq \mathbb {R} ^{n}}$ instead of Lebesgue measure, as proven in sections 2.8–2.9 of Federer's Geometric Measure Theory, 1969.