Lebesgue density theorem
where denotes the ball of radius centered at .
The Lebesgue density theorem asserts that for almost every point of the density
exists and is equal to .
In other words, for every measurable set the density of is or almost everywhere. However, it is a curious fact that if and , then there are always points of where the density is neither nor [1, Lemma 4].
- 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=0499.10035Zbl 0499.10035.