Absolute Measurable Spaces (Encyclopedia of Mathematics and by Togo Nishiura

By Togo Nishiura

Absolute measurable area and absolute null house are very previous topological notions, constructed from recognized evidence of descriptive set conception, topology, Borel degree concept and research. This monograph systematically develops and returns to the topological and geometrical origins of those notions. Motivating the improvement of the exposition are the motion of the crowd 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 area, extension of the Purves theorem and up to date advances on homeomorphic Borel likelihood measures at the Cantor house, are among the subject matters mentioned. A short dialogue of set-theoretic effects on absolute null area 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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Let us begin with the statement of the partition theorem. This theorem is a purely set theoretic one; that is, there are no topological assumptions made. Also the continuum hypothesis is not required. For the reader’s benefit, we shall include also the beautiful proof in [120]. 37. Let X be a set with card(X ) = ℵ1 , and let K be a class of subsets of X with the following properties: (1) K is a σ -ideal, (2) the union of K is X , (3) K has a subclass G with card(G) = ℵ1 and the property that each member of K is contained in some member of G, (4) the complement of each member of K contains a set with cardinality ℵ1 that belongs to K.

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 .

It is well-known that every separable metrizable space can be topologically embedded into the Hilbert cube [0, 1] N . Also, there is a B-homeomorphism ϕ of the Hilbert cube onto {0, 1}N , which is homeomorphic to the classical Cantor ternary set. Consequently, the study of univ M(X ) only from the point of view of universally measurable sets in X can be carried out on subspaces of the Cantor set. The difficulty is that the B-homeomorphism ϕ does not preserve the topological structure of the space X .

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