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| In [[mathematics]], '''Lebesgue's density theorem''' states that for any [[Lebesgue measure|Lebesgue measurable set]] <math>A\subset \R^n</math> , 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 [[null set|negligible]].
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| Let μ be the Lebesgue measure on the [[Euclidean space]] '''R'''<sup>''n''</sup> and ''A'' be a Lebesgue measurable subset of '''R'''<sup>''n''</sup>. Define the '''approximate density''' of ''A'' in a ε-neighborhood of a point ''x'' in '''R'''<sup>''n''</sup> as
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| :<math> d_\varepsilon(x)=\frac{\mu(A\cap B_\varepsilon(x))}{\mu(B_\varepsilon(x))}</math> | |
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| where ''B''<sub>ε</sub> denotes the [[closed ball]] of radius ε centered at ''x''.
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| '''Lebesgue's density theorem''' asserts that for almost every point ''x'' of ''A'' the '''density'''
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| :<math> d(x)=\lim_{\varepsilon\to 0} d_{\varepsilon}(x)</math>
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| exists and is equal to 1.
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| In other words, for every measurable set ''A'', the density of ''A'' is 0 or 1 [[almost everywhere]] in '''R'''<sup>''n''</sup>.<ref>{{cite book| last = Mattila| first = Pertti| title = Geometry of Sets and Measures in Euclidean Spaces: Fractals and Rectifiability| year = 1999| isbn = 978-0-521-65595-8 }}</ref> However, it is a curious fact that if μ(''A'') > 0 and {{nowrap|μ('''R'''<sup>''n''</sup> \ ''A'') > 0}}, then there are always points of '''R'''<sup>''n''</sup> where the density is neither 0 nor 1.
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| 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.
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| The Lebesgue density theorem is a particular case of the [[Lebesgue differentiation theorem]].
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| == See also == | |
| * [[Boundary (topology)]], analog in topology
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| == References ==
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| {{reflist}}
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| * Hallard T. Croft. Three lattice-point problems of Steinhaus. ''Quart. J. Math. Oxford (2)'', 33:71-83, 1982.
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| {{PlanetMath attribution|id=3869|title=Lebesgue density theorem}}
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| [[Category:Theorems in measure theory]]
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| [[Category:Integral calculus]]
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