Download Absolute Measurable Spaces (Encyclopedia of Mathematics and by Togo Nishiura PDF

By Togo Nishiura

Absolute measurable area and absolute null house are very previous topological notions, constructed from famous evidence of descriptive set thought, 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. life of uncountable absolute null house, extension of the Purves theorem and up to date advances on homeomorphic Borel likelihood measures at the Cantor area, 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 concept.

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Example text

43. The cardinal numbers κG and κ0 are equal. Proof. Let S ⊂ {0, 1}N and µ ∈ M {0, 1}N with µ∗ (S) > 0. Then the restricted measure ν = µ|S and A = B(S) will result in a measurable separable σ -algebra A on the set S, whence κG ≤ κ0 . To prove the other inequality, let µ be a nontrivial measure on a separable σ -algebra A on a set S. A topology τ on the set S corresponding to this separable σ -algebra A results in a topological embedding f of S into {0, 1}N . 4. Grzegorek’s cardinal number κG 21 ν = f# µ is a nontrivial, continuous, complete, finite Borel measure on {0, 1}N and ν ∗ f [S] = µ(S) > 0.

As M ⊂ X we have V ∩M = M \FX (M ) ∈ abNULL. Hence Y \ V ⊃ FY (M ). The proposition follows because FX (M ) = X ∩ (Y \ V ) ⊃ ✷ X ∩ FY (M ). Another property is the topological invariance of the positive closure operator. 2. 13. For homeomorphisms h : X → Y of separable metrizable spaces X and Y , if M ⊂ X , then FY (h[M ]) = h[FX (M )]. Proof. Denote the open subset X \ FX (M )] of X by U . Then U ∩ M is a universally null set in X . It follows that h[U ∩ M ] is a universally null set in Y . As h[U ∩ M ] = h[U ] ∩ h[M ] and h[U ] is an open set in Y , we have h[U ∩ M ] ⊂ Y \ FY (h[M ]), whence FY (h[M ]) ⊂ h[M ∩ FX (M )].

So C = X \ f −1 [U ( f )] is an absolute Borel space. 22. It was shown there that there is a continuous injection g : N → graph( f |C) such that graph( f |C) \ g[N ] is a countable set. With the natural projection π : graph( f |C) → Y , the composition h = πg provides a collection Bn , n = 1, 2, . . , of Borel subsets of N such that N = ∞ n=1 Bn and h|Bn is a homeomorphism for each n. Note that Cn = C ∩ π1 g[Bn ], where π1 is the natural projection of graph( f |C) onto C, is an absolute Borel space and f |Cn is B-homeomorphism of Cn onto h[Bn ].

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