Topology

Absolute Measurable Spaces (Encyclopedia of Mathematics and by Togo Nishiura

By Togo Nishiura

Absolute measurable area and absolute null house are very outdated topological notions, built from famous evidence of descriptive set thought, topology, Borel degree concept and research.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

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As X is an absolute measurable space there is an absolute Borel space B contained in X with µ(X \ B) = 0. So B is uncountable. Let Y be a topological copy of {0, 1}N contained in B and define B0 = (X \ Y ) ∪ B0 and Bα = Bα for 0 < α < ω1 , where Bα , α < ω1 is a transfinite sequence that is m-convergent in Y . It is easily seen that Bα , α < ω1 , is m-convergent in X . 28). ✷ This characterization theorem has been proved without the aid of the continuum hypothesis. 36. There exist absolute null spaces of cardinality ℵ1 .

The same is true for the graph of f , which will be shown next. 22. If f : X → Y is a universally measurable map, where X and Y are separable metrizable spaces, then graph( f ) is a universally measurable set in X × Y . 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.

For a subset S of {0, 1}N and a µ in M {0, 1}N , the outer measure µ∗ (S) is a well defined nonnegative real number. Define the cardinal number κ0 as follows: κ0 = min card(S) : µ∗ (S) > 0 for some µ in M {0, 1}N . We prove the following lemma due to Grzegorek [68]. 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 .

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