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a-field a-finite affine subspace analytic metric space analytic sets arbitrary balls Borel measure Borel outer measure Borel set called choose clearly closed compact metric space compact set compact subsets consider construction containing covering cubes decreasingly continuous define definition denote the paving diam diameter e-packing Euclidean spaces exist f(diam finite dimensional finitely founded geometric measure half-open intervals Hausdorff dimension Hausdorff measure hence Hf(A homeomorphism inequality ip,K ip(A ip(B ip(K k-dimensional surface element Lebesgue integrals Lebesgue measure Lemma let f Let ip Let T,T Let X,d metric additive metric outer measure metric space X,d non-empty notation obtain open set open subset packing measure pairwise disjoint Polish space positively separated premeasure prewellordering prove PT(K separable metric space sequence set function Souslin space and let subadditive Theorem tight and finitely topology trivially type-1 measure induced write