Lebesgue's density theorem
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In mathematics , Lebesgue's density theorem states that for any Lebesgue measurable set , the "density" of A is 0 or 1 at almost every point in . 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 R n and A be a Lebesgue measurable subset of R n . Define the approximate density of A in a ε-neighborhood of a point x in R n as
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 A the density
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 R n . [1] However, if μ( A ) > 0 and μ( R n \ A ) > 0 , then there are always points of R n where the density 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 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 R n instead of Lebesgue measure, see Discussion .
See also
- Lebesgue differentiation theorem – Mathematical theorem in real analysis
References
- ↑ Mattila, Pertti (1999). Geometry of Sets and Measures in Euclidean Spaces: Fractals and Rectifiability . ISBN 978-0-521-65595-8 .
- Hallard T. Croft. Three lattice-point problems of Steinhaus. Quart. J. Math. Oxford (2) , 33:71-83, 1982.
This article incorporates material from Lebesgue density theorem on PlanetMath , which is licensed under the Creative Commons Attribution/Share-Alike License .
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