It often occurs that a Borel finite measure plays an important role in estimating the Hausdorff dimension of a Borel set of the Euclidian space, for example [6, 12, 16, 21, 22]. In particular, in the Euclidean space and homogeneous (in the sense of Coifman and Weiss) space, Frostman's theorem [8, 14] and Tricot's theorem [20] hold and those facts imply the equivalence among dimensions and several indices of a finite Borel measure [7]. But it is not known whether Frostman's theorem and Tricot's theorem can be extended to general metric space or not.
The aim of this paper is to analyse systematically the several fractional dimensions of a subset and a finite Borel measure and establish new relations among them in a general separable metric space. As an application, we prove that Frostman's and Tricot's theorems are equivalent.