By Togo Nishiura

Absolute measurable house and absolute null area are very outdated topological notions, built from recognized evidence of descriptive set concept, topology, Borel degree conception and research. This monograph systematically develops and returns to the topological and geometrical origins of those notions. Motivating the advance of the exposition are the motion of the gang of homeomorphisms of an area on Borel measures, the Oxtoby-Ulam theorem on Lebesgue-like measures at the unit dice, and the extensions of this theorem to many different topological areas. lifestyles of uncountable absolute null house, extension of the Purves theorem and up to date advances on homeomorphic Borel likelihood measures at the Cantor house, are among the issues mentioned. A short dialogue of set-theoretic effects on absolute null house is given, and a four-part appendix aids the reader with topological measurement concept, Hausdorff degree and Hausdorff measurement, and geometric degree thought.

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Universally positive closure. Let M be a subset of a separable metrizable space X . Denote by V the collection of all open sets V such that V ∩ M is a universally null set in X . As X is a Lindeloff space, there is a countable sub∞ V = collection V0 , V1 , . . , of V such that V = i=0 Vi . Since univ N(X ) is a σ -ideal, we have V ∩ M is a universally null set in X . We call the closed set FX (M ) = X \V the universally positive closure of M in X (or, positive closure for short). FX is not the topological closure operator ClX , but it does have the following properties.

Since univ N(X ) is a σ -ideal, we have V ∩ M is a universally null set in X . We call the closed set FX (M ) = X \V the universally positive closure of M in X (or, positive closure for short). FX is not the topological closure operator ClX , but it does have the following properties. 11. Let X be a separable metrizable space. Then the following statements hold. (1) If M1 ⊂ M2 ⊂ X , then FX (M1 ) ⊂ FX (M2 ). (2) If M1 and M2 are subsets of X , then FX (M1 ∪ M2 ) = FX (M1 ) ∪ FX (M2 ). (3) If M ⊂ X , then FX (M ) = FX M ∩ FX (M ) = ClX M ∩ FX (M ) .

We may assume µn Un ∩ F(X ) < 2−n . Let νn = µn Un ∩F(X ) for each n. Then, for each Borel set B, we have ν(B) = ∞ n=0 νn (B) < 2. Also, ν({x}) = 0 for every point x of X . Hence ν determines a continuous, complete, finite Borel measure on X . We already know support(ν) ⊂ F(X ). Let U be an open set such that U ∩ F(X ) = ∅. There exists an n ✷ such that U ⊃ Un ∩ F(X ) = ∅, whence ν(U ) > 0. Hence F(X ) ⊂ support(ν). 15. Let X be a separable metrizable space. If M is a subset of X with FX (M ) = ∅, then support(µ) = FX (M ) for some continuous, complete, finite Borel measure µ on X .