By Togo Nishiura

Absolute measurable area and absolute null house are very previous topological notions, built from famous proof of descriptive set thought, topology, Borel degree concept 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 among the themes mentioned. A short dialogue of set-theoretic effects on absolute null house is given, and a four-part appendix aids the reader with topological measurement concept, Hausdorff degree and Hausdorff measurement, and geometric degree conception.

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There exists an n ✷ such that U ⊃ Un ∩ F(X ) = ∅, whence ν(U ) > 0. Hence F(X ) ⊂ support(ν). 15. Let X be a separable metrizable space. If M is a subset of X with FX (M ) = ∅, then support(µ) = FX (M ) for some continuous, complete, finite Borel measure µ on X . Proof. We have FM (M ) = M ∩ FX (M ) = ∅. Hence there is a measure ν in MEASpos (M ). Let µ be an extension of ν such that support(µ) = FX (M ). Such an extension will exist with the aid of the inclusion map of FM (M ) into X . ✷ Of course, there are spaces for which the existence of positive measures is obvious – for example, the unit n-cube [0, 1]n has the Lebesgue measure.

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.

Let { xα : α < η } be a well ordering of X by an ordinal number η. This well ordering corresponds to a unique subset R of X × X . As every subset of X × X is an absolute measurable space we have that R is an absolute measurable space that is contained in Y × Y such that X is well ordered by R. Recaw’s theorem is the following. 52 (Recaw). Let R be an absolute measurable space contained in [0, 1] × [0, 1]. Then, any subset X of [0, 1] that is well ordered by the relation R is an absolute null space contained in [0, 1].