Absolute Measurable Spaces (Encyclopedia of Mathematics and by Togo Nishiura

By Togo Nishiura

Absolute measurable house and absolute null area are very previous topological notions, built from famous proof of descriptive set concept, 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 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 house, 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 conception, Hausdorff degree and Hausdorff measurement, and geometric degree conception.

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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.

Darst [37]. A “converse” statement will be investigated in the next chapter on universally measurable sets in a fixed space X , namely, we shall present the theorem due to Darst and Grzegorek. 51. Let f : X → Y be a B-map, where X is an absolute Borel space. Then f [M ] is an absolute null space whenever M is an absolute null space contained in X . Proof. Let us prove that every subset of f [M ] is an absolute measurable space. To this end, let Y0 ⊂ f [M ]. Note that M0 = M ∩ f −1 [Y0 ] is an absolute measurable space since M is an absolute null space.

3. 5. Existence under the continuum hypothesis. There is a novel presentation of the existence of absolute null spaces with the aid of the continuum hypothesis in the book Measure and Category by J. C. Oxtoby [120]. 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].

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