Convergence and Concentration of Metrics and Measures

When we speak of measures μ on a metric space X, we always assume that X is a Polish space, i.e., complete with a countable base, and that “measure” means a Borel measure, where all Borel subsets in X are measurable. We are mostly concerned with finite measures where X has finite (total) mass μ-(X) < ∞, but we also allow σ-finite measure spaces X, which are the countable unions of Xi with μ(Xi) < ∞. For example, the ordinary n dimensional Hausdorff measure in ℝn is σ-finite, while the k-dimensional Hausdorff measure on ℝn for k < n is not σ-finite. But, we may restrict such a measure to a k-dimensional submanifold V ⊂ ℝn, i.e., we declare μk(U) = μk(U ∩ V) for all open U ⊂ ℝn, in which case the measure becomes σ-finite and admissible in our discussion.