Absolute Measurable Spaces (Encyclopedia of Mathematics and by Togo Nishiura

By Togo Nishiura

Absolute measurable area and absolute null area are very previous topological notions, constructed from recognized proof of descriptive set idea, topology, Borel degree conception 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 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. life of uncountable absolute null area, extension of the Purves theorem and up to date advances on homeomorphic Borel chance measures at the Cantor area, are among 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 size concept, Hausdorff degree and Hausdorff size, and geometric degree conception.

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Let A be a Borel set in [0, 1] N such that A ⊃ X0 and such that the inner measure µ∗ (A \ X0 ) is 0. 5. More on existence of absolute null spaces 25 { (x, y) ∈ A×A : y R x }. As W = (A×A)∩R−1 we have that W is an absolute measurable space. Hence there is a Borel set V such that V ⊂ W and µ × µ (W \ V ) = 0. As usual, we denote { y ∈ [0, 1] N : (x, y) ∈ V } by Vx , a Borel set in [0, 1] N . As the function x → µ(Vx ) is Borel measurable, the set { x : µ(Vx ) > 0 } is a Borel subset of [0, 1] N .

Clearly f [M0 ] = Y0 , whence an absolute measurable space. 20. 5. More on existence of absolute null spaces We have seen earlier that Hausdorff’s m-convergence, a sufficient condition for the existence of uncountable absolute null spaces, was essentially a part of a characterization theorem. 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.

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. Clearly f [M0 ] = Y0 , whence an absolute measurable space. 20. 5. More on existence of absolute null spaces We have seen earlier that Hausdorff’s m-convergence, a sufficient condition for the existence of uncountable absolute null spaces, was essentially a part of a characterization theorem. The characterization used a well ordering of ℵ1 -many disjoint subsets of an uncountable absolute measurable space.

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