By Togo Nishiura

ISBN-10: 0521875560

ISBN-13: 9780521875561

Absolute measurable area and absolute null area are very outdated topological notions, built from famous proof of descriptive set idea, topology, Borel degree thought and research.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

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**Sample text**

Since every metric space is metrizable, this change in definition will cause no loss in the analysis of any specific metric space. The next chapter will be concerned with the notion of universally measurable sets X in a metrizable space Y . This notion is the appropriate modification of the property 12 An extended discussion of Shortt’s observation can be found in Appendix B. 6. Coments 27 M(rel Y ). The modifier universally is used to indicate that the metrizable space Y is fixed, not the metric.

The characterization used a well ordering of ℵ1 -many disjoint subsets of an uncountable absolute measurable space. The well ordering idea was used to advantage by Recaw [130] for subspaces of R and by Plewik [127, Lemma] for subspaces of {0, 1}N to prove another sufficient condition for the existence of absolute null spaces. It was pointed out by Recaw that, with the aid of B-homeomorphisms, Rn can replace the ambient space R. Of course the use of B-homeomorphisms defeats the emphasis on topological homeomorphism if one can avoid the use of B-homeomorphisms.

54. Let Y be a separable metrizable space. In order that a subspace X of Y be an absolute null space it is necessary and sufficient that there exists an absolute measurable space R contained in Y × Y with the property that X is well ordered by the relation R. 52 generalizes a result of Plewik [127, Lemma ] who assumes that the relation R is formed by a Borel subset of [0, 1] × [0, 1] – clearly R is an absolute measurable space. Actually, Plewik works in the collection P(ω) = { X : X ⊂ ω }, which is homeomorphic to the space {0, 1}N .

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

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