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单词 LebesgueDensityTheorem
释义

Lebesgue density theorem


Let μ be the Lebesgue measureMathworldPlanetmath on n, and for ameasurable setMathworldPlanetmath An define the density of A inϵ-neighborhoodMathworldPlanetmath of xn by

dϵ(x)=μ(ABϵ(x))μ(Bϵ(x))

where Bϵ(x) denotes the ball of radius ϵcentered at x.

The Lebesgue density theorem asserts that for almost every point ofA the density

d(x)=limϵ0dϵ(x)

exists and is equal to 1.

In other words, for every measurable set A the density of A is0 or 1 almost everywhere. However, it is a curious fact thatif μ(A)>0 and μ(nA)>0, then there arealways points of n where the density is neither 0nor 1 [1, Lemma 4].

References

  • 1 Hallard T. Croft. Three lattice-point problems of Steinhaus. Quart. J. Math. Oxford (2), 33:71–83, 1982. http://www.emis.de/cgi-bin/zmen/ZMATH/en/quick.html?type=html&an=0499.10035Zbl0499.10035.
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