{{Short description|Theorem in analysis}} In mathematics, '''Lebesgue's density theorem''' states that for any Lebesgue measurable set <math>A\subseteq \R^n</math>, the "density" of <math>A</math> is 0 or 1 at almost every point in <math>\R^n</math>. Additionally, the "density" of <math>A</math> is 1 at almost every point of <math>A</math>. Intuitively, this means that the boundary of <math>A</math>, the set of points in <math>A</math> for which all neighborhoods are partially in <math>A</math> and partially outside <math>A</math>, is of measure zero.
right|thumb|400px|alt=Lebesgue's density theorem, applied to the inside of a square, its corners, edges, inside, and outside|Lebesgue's density theorem, applied to the inside of a square, its corners, edges, inside, and outside
==The definition==
Let <math>\mu</math> be the Lebesgue measure on the Euclidean space and <math>A\subseteq \R^n</math> be a Lebesgue measurable set. Let <math>x\in \R^n</math> and let <math>B</math><sub>ε</sub><math>(x)</math> denote the open ball of radius <math>\varepsilon</math> centered at <math>x</math>. Define
:<math>\qquad\qquad d_\varepsilon(x)=\frac{\mu(A\cap B_\varepsilon(x))}{\mu(B_\varepsilon(x))}</math>
'''Lebesgue's density theorem''' asserts that for almost every point <math>x</math> of <math>A\subseteq \R^n</math> the '''density'''
:<math>\qquad\qquad\qquad d(x)=\lim_{\varepsilon\to 0} d_{\varepsilon}(x)</math>
exists and is equal to 0 or 1.
==What the Lebesgues density theorem states==
For every measurable set <math>A</math>, the density of <math>A</math> is 0 or 1 almost everywhere<ref>{{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 }}</ref>. If <math>0<\mu(A)<\infty</math>, then there are always points of <math>A\subseteq \R^n</math> where the density either does not exist or exists but is neither 0 nor 1.<ref>{{cite journal | last = Croft | first = Hallard |authorlink=Hallard Croft | title = Three lattice-point problems of Steinhaus | journal = Quarterly J. Math. Oxford (2)| volume = 33 | pages= 71–83 | year=1982 }}</ref>.
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 <math>A\subseteq \R^n</math> instead of Lebesgue measure, as proven in sections 2.8–2.9 of Federer's ''Geometric Measure Theory'', 1969.
==See also==
* {{annotated link|Lebesgue differentiation theorem}}
==References== {{reflist}}
{{PlanetMath attribution|id=3869|title=Lebesgue density theorem}}
{{Measure theory}}
Category:Theorems in measure theory Category:Integral calculus