Absolute Measurable Spaces (Encyclopedia of Mathematics and by Togo Nishiura

By Togo Nishiura

Absolute measurable house and absolute null area are very outdated topological notions, constructed from recognized evidence of descriptive set thought, topology, Borel degree thought and research.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 area, extension of the Purves theorem and up to date advances on homeomorphic Borel likelihood measures at the Cantor house, are one of the issues 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 idea, Hausdorff degree and Hausdorff measurement, and geometric degree thought.

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4. Grzegorek’s cardinal number κG It is easily seen that uncountable absolute measurable spaces that are not absolute null spaces must have cardinality c = 2ℵ0 . The question of the existence of a non absolute null space X that has the same cardinality as some uncountable absolute null space X is addressed in this section. This is a question of S. 4. Grzegorek’s cardinal number κG 19 footnote 10 after all the required definitions are given. We shall present an example, due to Grzegorek [68], which gives an affirmative answer.

Assume the continuum hypothesis. If X is an uncountable, separable completely metrizable space that contains no isolated points, then there exists an uncountable absolute null space contained in X . Observe that every uncountable absolute Borel space contains a topological copy of the Cantor space. Hence the continuum hypothesis implies the existence of an uncountable absolute null space contained in each uncountable completely metrizable space. We remind the reader that it was the novelty of the partition theorem that motivated this section on the continuum hypothesis.

45. There exists a subspace X of {0, 1}N such that X is an absolute null space with card(X ) = κG . 11 A cardinal number κ will be identified with the minimal initial segment of ordinal numbers whose cardinal number is κ. 21 we have the following corollary, where P(X ) is the power set of X . 46. If (X , µ, P(X )) is a nonnegative real-valued, continuous measure space with card(X ) = κG , then µ is identically equal to 0. 3. abNULL is not preserved by Borel measurable maps. Let us begin with a simple proposition.

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