Absolute Measurable Spaces (Encyclopedia of Mathematics and by Togo Nishiura

By Togo Nishiura

Absolute measurable area and absolute null area are very outdated topological notions, built from famous evidence of descriptive set thought, topology, Borel degree thought and research.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 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. life of uncountable absolute null house, extension of the Purves theorem and up to date advances on homeomorphic Borel chance measures at the Cantor house, are one of many issues 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 size, and geometric degree thought.

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Extra info for Absolute Measurable Spaces (Encyclopedia of Mathematics and its Applications)

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50. 58 gives condition (4) implies condition (5). Obviously, condition (5) implies condition (6). It remains to prove that condition (6) implies condition (2). Suppose that condition (2) fails. 49, condition (6) fails. This completes the proof. 3. 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 , .

Moreover, univ N(X ) = univ M(X ) = P(X ). Assume F(X ) = ∅. Let us prove that univ M(X ) ⊃ univ Mpos (X ). To this end, let M ∈ univ Mpos (X ) and let µ be a continuous, complete, finite Borel measure on X . 3. 16 there is a positive, continuous, complete, finite Borel measure ν on X such that µ support(µ) = ν support(µ) . Since M ∈ M(X , ν), we have M ∩ support(µ) ∈ M(X , ν); and from M ∩ support(µ) ⊂ support(µ) we conclude M ∩ support(µ) ∈ M(X , µ). Also, from the completeness of µ, we have M \ support(µ) ∈ N(X , µ).

Additionally, if X is an absolute measurable space, then so is graph( f ). Proof. Let µ be a complete, finite Borel measure on X × Y and denote the natural projection of X × Y onto X by π. As ν = π# µ is a complete, finite Borel measure on X , there is a Borel measurable map g : X → Y such that E = { x : f (x) = g(x) } has ν measure equal to 0. Let A be a Borel set in X such that E ⊂ A and ν(A) = 0. We have that graph( f ) = π −1 [A] ∩ graph( f ) ∪ π −1 [X \ A] ∩ graph(g) . As the first summand has µ measure equal to 0 and the second summand is a Borel set in X × Y , we have that graph( f ) is µ-measurable.

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