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Absolute measurable area and absolute null area are very outdated topological notions, constructed from recognized evidence of descriptive set idea, topology, Borel degree idea and research. This monograph systematically develops and returns to the topological and geometrical origins of those notions. Motivating the advance 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 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 size conception, Hausdorff degree and Hausdorff size, and geometric degree idea.

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35). 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.

44 (Grzegorek). Let S ⊂ {0, 1}N and let µ be a continuous, complete, finite Borel measure on {0, 1}N such that card(S) = κ0 and µ∗ (S) > 0. Then there exists a nonmeasurable separable σ -algebra A on S. Proof. Let ν = µ|S be the nontrivial continuous measure on the σ -algebra B(S) of all Borel subsets of the topological space S and let B = { Ui : i < ω0 } be a countable base for the open subsets of S. Let sα , α ∈ κG ,11 be a well ordering of the set S. For each α let Gα be an open subset of the metrizable space S such that { sβ : β < α } ⊂ Gα and ν(Gα ) ≤ 12 ν(S).

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 card( f [S]) = card(S), we have κ0 ≤ card(S). Therefore, κ0 ≤ κG and the lemma is proved. ✷ We now give Grzegorek’s theorem. 44 (Grzegorek). Let S ⊂ {0, 1}N and let µ be a continuous, complete, finite Borel measure on {0, 1}N such that card(S) = κ0 and µ∗ (S) > 0. Then there exists a nonmeasurable separable σ -algebra A on S. Proof. Let ν = µ|S be the nontrivial continuous measure on the σ -algebra B(S) of all Borel subsets of the topological space S and let B = { Ui : i < ω0 } be a countable base for the open subsets of S.

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