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- Lebesgue's_density_theorem abstract "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 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 aswhere Bε denotes the closed ball of radius ε centered at x. Lebesgue's density theorem asserts that for almost every point x of A the densityexists and is equal to 1.In other words, for every measurable set A, the density of A is 0 or 1 almost everywhere in Rn. However, it is a curious fact that if μ(A) > 0 and μ(Rn \ A) > 0, then there are always points of Rn where the density is neither 0 nor 1[citation needed].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.".
- Lebesgue's_density_theorem wikiPageID "4258134".
- Lebesgue's_density_theorem wikiPageRevisionID "598866258".
- Lebesgue's_density_theorem hasPhotoCollection Lebesgue's_density_theorem.
- Lebesgue's_density_theorem id "3869".
- Lebesgue's_density_theorem title "Lebesgue density theorem".
- Lebesgue's_density_theorem subject Category:Integral_calculus.
- Lebesgue's_density_theorem subject Category:Theorems_in_measure_theory.
- Lebesgue's_density_theorem type Abstraction100002137.
- Lebesgue's_density_theorem type Communication100033020.
- Lebesgue's_density_theorem type Message106598915.
- Lebesgue's_density_theorem type Proposition106750804.
- Lebesgue's_density_theorem type Statement106722453.
- Lebesgue's_density_theorem type Theorem106752293.
- Lebesgue's_density_theorem type TheoremsInMeasureTheory.
- Lebesgue's_density_theorem comment "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 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.".
- Lebesgue's_density_theorem label "Dichtheidsstelling van Lebesgue".
- Lebesgue's_density_theorem label "Lebesgue's density theorem".
- Lebesgue's_density_theorem label "Teorema di densità di Lebesgue".
- Lebesgue's_density_theorem label "Twierdzenie Lebesgue’a o punktach gęstości".
- Lebesgue's_density_theorem label "ルベーグの密度定理".
- Lebesgue's_density_theorem sameAs Teorema_di_densità_di_Lebesgue.
- Lebesgue's_density_theorem sameAs ルベーグの密度定理.
- Lebesgue's_density_theorem sameAs 르베그_밀도_정리.
- Lebesgue's_density_theorem sameAs Dichtheidsstelling_van_Lebesgue.
- Lebesgue's_density_theorem sameAs Twierdzenie_Lebesgue’a_o_punktach_gęstości.
- Lebesgue's_density_theorem sameAs m.0bsr_w.
- Lebesgue's_density_theorem sameAs Q842953.
- Lebesgue's_density_theorem sameAs Q842953.
- Lebesgue's_density_theorem sameAs Lebesgue's_density_theorem.
- Lebesgue's_density_theorem wasDerivedFrom Lebesgue's_density_theorem?oldid=598866258.
- Lebesgue's_density_theorem isPrimaryTopicOf Lebesgue's_density_theorem.