Absolute Measurable Spaces (Encyclopedia of Mathematics and by Togo Nishiura

By Togo Nishiura

Absolute measurable house and absolute null area are very outdated topological notions, constructed from recognized proof of descriptive set thought, topology, Borel degree conception and research.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. life of uncountable absolute null area, extension of the Purves theorem and up to date advances on homeomorphic Borel chance measures at the Cantor house, are one of the themes 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 thought, Hausdorff degree and Hausdorff size, and geometric degree idea.

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It is natural to want to consider functions f : X → R in the context of absolute measurable spaces, where X is a separable, metrizable space. Given such a space X , we define the σ -ring ab M(X ) = { M : M ⊂ X , M ∈ abMEAS }. This σ -ring is a σ -algebra if and only if X ∈ abMEAS. It is natural to say that a function f : X → R is absolutely measurable if M ∩ f −1 [F] is in ab M(X ) for every closed set F of R and every M in ab M(X ). Observe, for an absolutely measurable f : X → R, that f −1 [F] ∈ ab M(X ) for every closed set F if and only if X ∈ ab M(X ).

34 (Sierpi´nski–Szpilrajn). Every co-analytic space X that is not an analytic space has a transfinite sequence Bα , α < ω1 , in abBOR that is m-convergent in X . Proof. 1) on page 181 of Appendix A, we have X =Y \A= α<ω1 Aα , where Y is some separable completely metrizable space and A is an analytic space. 9, we have that the constituents Aα are absolute Borel spaces and that the collection of those constituents which are nonempty is uncountable. Hence there is a natural transfinite subsequence Bα , α < ω1 , of Borel sets such that the first three conditions of the definition of 16 The absolute property m-convergence in X are satisfied.

Let M be a subset of a separable metrizable space X . Then M ∈ univ N(X ) if and only if M is an absolute null space (that is, M ∈ abNULL). Proof. It is clear that M ∈ univ N(X ) whenever M ∈ abNULL and M ⊂ X . So let M ∈ univ N(X ). 20 that M ∈ abNULL. 8. For separable metrizable spaces X and Y , let f be a Bhomeomorphism of X onto Y . Then, for subsets M of X , (1) f −1 [M ] ∈ B(X ) if and only if M ∈ B(Y ), (2) f −1 [M ] ∈ univ M(X ) if and only if M ∈ univ M(Y ), (3) f −1 [M ] ∈ univ N(X ) if and only if M ∈ univ N(Y ).

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