By Togo Nishiura

Absolute measurable area and absolute null house are very previous topological notions, constructed from recognized evidence 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 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 house, are among the themes 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 measurement, and geometric degree concept.

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Let Y be a separable metrizable space. If M ⊂ X ⊂ Y , then FX (M ) = X ∩ FY (M ). Proof. Observe that (Y \FY (M ))∩M ∈ abNULL. As M ⊂ X we have (Y \FY (M ))∩ M = (X \ FY (M )) ∩ M . So, FX (M ) ⊂ X ∩ FY (M ). Next let V be an open subset of Y such that V ∩X = X \FX (M ). As M ⊂ X we have V ∩M = M \FX (M ) ∈ abNULL. Hence Y \ V ⊃ FY (M ). The proposition follows because FX (M ) = X ∩ (Y \ V ) ⊃ ✷ X ∩ FY (M ). Another property is the topological invariance of the positive closure operator. 2. 13.

We have the following characterization. 17. Let X be a separable metrizable space. Then univ M(X ) = univ M pos (X ) and univ N(X ) = univ Npos (X ). This characterization will turn out to be quite useful in the investigation of the unit n-cube [0, 1]n . We shall see later in this chapter that the homeomorphism group of the space [0, 1] will play a nice role in a characterization of univ M [0, 1] . Proof of Theorem. If F(X ) = ∅, then MEASpos (X ) = ∅ and X ∈ abNULL. Hence univ Mpos (X ) = ∅ which, by the usual convention, is equal to P(X ) = { E : E ⊂ X }.

From the decomposition, which is assured by the above lemma, select exactly one point from each Xα to form the set X . Let K be any set of the first category of Baire. There is a β with β < ω1 and K ⊂ α≤β Xα . Hence, card(K ∩ X ) ≤ ℵ0 . ✷ Finally we have need of the following lemma which will be left as an exercise. 40. Let X be an uncountable, separable completely metrizable space and let M(X , µ) be a continuous, complete, σ -finite Borel measure space on X . If D is a countable dense subset of X , then there exists a Gδ subset E of X that contains D such that µ(E) = 0 and X \ E is an uncountable Fσ subset of X of the first category of Baire.