mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...

, Lebesgue's density theorem states that for any Lebesgue measurable set

A\subset \R^n

, the "density" of ''A'' is 0 or 1 at

almost every In measure theory (a branch of mathematical analysis), a property holds almost everywhere if, in a technical sense, the set for which the property holds takes up nearly all possibilities. The notion of "almost everywhere" is a companion notion t ...

point in

\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 Euclidean space is the fundamental space of geometry, intended to represent physical space. Originally, that is, in Euclid's Elements, Euclid's ''Elements'', it was the three-dimensional space of Euclidean geometry, but in modern mathematics ther ...

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 :

d_\varepsilon(x)=\frac

where ''B''_ε denotes the

closed ball In mathematics, a ball is the solid figure bounded by a ''sphere''; it is also called a solid sphere. It may be a closed ball (including the boundary points that constitute the sphere) or an open ball (excluding them). These concepts are defin ...

of radius ε centered at ''x''. Lebesgue's density theorem asserts that for almost every point ''x'' of ''A'' the density :

d(x)=\lim_ d_(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 measure theory (a branch of mathematical analysis), a property holds almost everywhere if, in a technical sense, the set for which the property holds takes up nearly all possibilities. The notion of "almost everywhere" is a companion notion to ...

in R^''n''. However, if μ(''A'') > 0 and , 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 In mathematics, the Lebesgue differentiation theorem is a theorem of real analysis, which states that for almost every point, the value of an integrable function is the limit of infinitesimal averages taken about the point. The theorem is named for ...

. Thus, this theorem is also true for every finite Borel measure on R^''n'' instead of Lebesgue measure, see

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References

* Hallard T. Croft. Three lattice-point problems of Steinhaus. ''Quart. J. Math. Oxford (2)'', 33:71-83, 1982. {{Measure theory Theorems in measure theory Integral calculus

See also

References