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Every complete doubling metric space carries a doubling measure. (English) Zbl 0897.28007

Summary: Nowadays a usual environment in analysis is a complete metric space equipped with a nontrivial Borel measure having a doubling property. It is known that a necessary condition for the existence of such a measure is that the space itself has a doubling property. In the present paper this condition is proved to be sufficient. More precisely, in Theorem 1 it is shown that if \(X\) is a complete metric space for which there exist constants \(C\geq 1\) and \(s\geq 0\) such that for all \(x\in X\), \(r>0\), and \(\lambda\geq 1\) the cardinality of every set in the closed ball \(B(x,\lambda r)\) whose points are at least \(r\) apart is at most \(C\lambda^s\), then for every \(t>s\) there exist a Borel measure \(\mu\) on \(X\) and a constant \(c=c(C,s,t)\geq 1\) such that \(0<\mu(B(x,\lambda r))\leq c\lambda^t\mu(B(x,r)) <\infty\) for all \(x,r,\lambda\) as above. In Theorem 2 every closed set \(X\subset{\mathbb R}^n\) is shown to carry such a measure \(\mu\) for \(t=n\) with \(c=c(n)\). These results are deduced from their special cases, due to A. L. Vol’berg and S. V. Konyagin [Izv. Akad. Nauk SSSR, Ser. Mat. 51, No. 3, 666-675 (Russian) (1987; Zbl 0649.42010); Math. USSR, Izv. 30, No. 3, 629-638 (English translation) (1988; Zbl 0727.28012)], where \(X\) is bounded and thus compact.

MSC:

28C15 Set functions and measures on topological spaces (regularity of measures, etc.)
54E45 Compact (locally compact) metric spaces
28A12 Contents, measures, outer measures, capacities
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