The Intrinsic Measure Theory of Riemannian and Euclidean Metric Spaces
Lynn H. Loomis · Annals of Mathematics · 1944
Euclidean spaces are boundedly compact and have the metric property that any two spheres are similar. In particular, if a closed sphere of radius r can be covered by n open spheres of radius x, then, for any positive a, every closed sphere of radius ar can be covered by n open spheres of radius ax. W\e shall show that this combinatorial similarity property is sufficient in any boundedly compact metric space for the development of ordinarv Lebesgue measure theorya. One result is the validity of the usual formula for the volume of a sphere. That is, apart from a multiplicative constant there is one and only one measure which is a volume in the sense that spheres of equal radii have equal measures, and there is an a such that the volume of a sphere of radius r is ra. The existence proof will be presented in a general enough form to include the development of the intrinsic measure theories of Riemannian metric spaces and of metric spaces like the Cantor sets for which the dimension a is non-integral.