In mathematics, Lebesgue's density theorem states that for any Lebesgue measurable set A ⊂ R n {\displaystyle A\subset \mathbb {R} ^{n}} , the "density" of A is 0 or 1 at almost every point in R n {\displaystyle \mathbb {R} ^{n}} . Additionally, the "density" of A is 1 at almost every point in A. Intuitively, this means that the "edge" of A, the set of points in A whose "neighborhood" is partially in A and partially outside of A, is negligible.

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

d ε ( x ) = μ ( A ∩ B ε ( x ) ) μ ( B ε ( x ) ) {\displaystyle d_{\varepsilon }(x)={\frac {\mu (A\cap B_{\varepsilon }(x))}{\mu (B_{\varepsilon }(x))}}}

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

d ( x ) = lim ε → 0 d ε ( x ) {\displaystyle d(x)=\lim _{\varepsilon \to 0}d_{\varepsilon }(x)}

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. However, if μ(A) > 0 and μ(Rn \ A) > 0, then there are always points of Rn where the density either does not exist or exists but is neither 0 nor 1 (, Lemma 4).

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 negligible.

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 Rn instead of Lebesgue measure, see Discussion.