Construction of Borel Measures on Metric Spaces |
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A E B A₁ analytic metric space analytic sets arbitrary assume B₁ balls Borel measure Borel outer measure Borel set called choose clearly closed compact set consider construction containing countable cubes d₁ decreasingly continuous define definition diam diameter Euclidean exist f(diam finite dimensional finitely founded H-dim H₁ H₂ half-open intervals Hausdorff dimension Hausdorff measure hence homeomorphism inequality IR¹ K₁ K₂ Lebesgue integrals Lebesgue measure Let x,d metric outer measure metric space X,d monotone notation o-finite obtain open set open subset P₁ packing measure pairwise disjoint Polish space positively separated premeasure prewellordering prove regular w.r.t. separable metric space sequence set function Souslin space and let subadditive T₁ Theorem tight and finitely topology trivially type-1 measure induced write x₁