Togo Nishiura's Absolute Measurable Spaces (Encyclopedia of Mathematics and PDF

By Togo Nishiura

ISBN-10: 0521875560

ISBN-13: 9780521875561

Absolute measurable area and absolute null house are very previous topological notions, constructed from recognized evidence of descriptive set conception, topology, Borel degree thought 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. 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 many subject matters 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 idea, Hausdorff degree and Hausdorff measurement, and geometric degree thought.

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Extra info for Absolute Measurable Spaces (Encyclopedia of Mathematics and its Applications)

Example text

So assume that X is not an absolute null space. Then there exists a continuous, complete, σ -finite Borel measure space M(X , µ) with µ(X ) > 0. 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 .

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.

6. Comments We close the chapter with a few comments. 1. Metric spaces. 2 are found in [152]. These early results concerning absolute measurable spaces were couched in the context of separable metric spaces, which differs from that of separable metrizable spaces used in our book. The definitions in the book are independent of the metric that can be assigned to a given metrizable space. Hence the notion of absolute measurable space as has been developed in the book is based solely on the Borel sets of a given separable, metrizable space, which are – of course – independent of the metrics that are associated with a given metrizable space.

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

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