By Togo Nishiura

ISBN-10: 0511721382

ISBN-13: 9780511721380

ISBN-10: 0521875560

ISBN-13: 9780521875561

Absolute measurable house and absolute null area are very outdated topological notions, built from famous proof of descriptive set concept, topology, Borel degree idea and research.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. lifestyles 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 one of several subject matters mentioned. A short dialogue of set-theoretic effects on absolute null area is given, and a four-part appendix aids the reader with topological size concept, Hausdorff degree and Hausdorff measurement, and geometric degree idea.

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**Example text**

We may assume µn Un ∩ F(X ) < 2−n . Let νn = µn Un ∩F(X ) for each n. Then, for each Borel set B, we have ν(B) = ∞ n=0 νn (B) < 2. Also, ν({x}) = 0 for every point x of X . Hence ν determines a continuous, complete, finite Borel measure on X . We already know support(ν) ⊂ F(X ). Let U be an open set such that U ∩ F(X ) = ∅. There exists an n ✷ such that U ⊃ Un ∩ F(X ) = ∅, whence ν(U ) > 0. Hence F(X ) ⊂ support(ν). 15. Let X be a separable metrizable space. If M is a subset of X with FX (M ) = ∅, then support(µ) = FX (M ) for some continuous, complete, finite Borel measure µ on X .

So, if E is a universally measurable set in X , then E is a (ν X )-measurable set in Y , whence a ν-measurable set in Y . ✷ The following theorems are essentially due to Sierpi´nski and Szpilrajn [142]. 7. 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 .

Recaw’s theorem is the following. 52 (Recaw). Let R be an absolute measurable space contained in [0, 1] × [0, 1]. Then, any subset X of [0, 1] that is well ordered by the relation R is an absolute null space contained in [0, 1]. We shall prove the following more general form. 53. Let Y be a separable metrizable space and let R be an absolute measurable space contained in Y × Y . Then, any subset X of Y that is well ordered by the relation R is an absolute null space contained in Y . Proof. There is no loss in assuming that Y is a subspace of the Hilbert cube [0, 1] N .

### Absolute Measurable Spaces (Encyclopedia of Mathematics and its Applications) by Togo Nishiura

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