Lebesgue's density theorem

{{Short description|Theorem in analysis}}

In mathematics, Lebesgue's density theorem states that for any Lebesgue measurable set $A\backslash subset\; \backslash R^n$, the "density" of A is 0 or 1 at almost every point in $\backslash R^n$. 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

:$d\_\backslash varepsilon(x)=\backslash frac\{\backslash mu(A\backslash cap\; B\_\backslash varepsilon(x))\}\{\backslash mu(B\_\backslash varepsilon(x))\}$

where B_ε 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

:$d(x)=\backslash lim\_\{\backslash varepsilon\backslash to\; 0\}\; d\_\{\backslash varepsilon\}(x)$

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.{{cite book| last = Mattila| first = Pertti|author-link = Pertti Mattila| title = Geometry of Sets and Measures in Euclidean Spaces: Fractals and Rectifiability| year = 1999| isbn = 978-0-521-65595-8 }} However, if μ(A) > 0 and {{nowrap|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 (,{{cite journal | last = Croft | first = Hallard | title = Three lattice-point problems of Steinhaus | journal = Quarterly J. Math. Oxford (2)| volume = 33 | pages= 71–83 | year=1982 }} 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.