By Togo Nishiura

Absolute measurable house and absolute null house are very outdated topological notions, built from famous proof of descriptive set idea, topology, Borel degree thought and research.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. 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 measurement conception, Hausdorff degree and Hausdorff size, and geometric degree idea.

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Let fα be such that fβ < fα < gα whenever β < α. The α-th step of the transfinite construction is now completed. Suppose that there is an h such that fα <* h <* gα for every α. Then for some m there will be uncountably many fβ such that δ( fβ , h) = m. Hence P(F<α , h) fails for some α, which contradicts ✷ the Hausdorff observation that P(F<α , h) holds whenever fα < h < gα . 3. The Sierpinski ´ and Szpilrajn example. Sierpi´nski and Szpilrajn gave this example in [142]. It uses the constituent decomposition of co-analytic spaces (see Appendix A, page 181).

Proof. Denote the inclusion map of X into Y by f . Let µ be a complete, finite Borel measure on X . Then f# µ is a complete, finite Borel measure on Y . Hence X is ( f# µ)measurable. It follows that the restriction measure ( f# µ)|X is the measure µ on the subspace X of Y . Consequently, if E is a universally measurable set in Y that is also a subset of X , then E is a universally measurable set in X . Next let ν be a complete, finite Borel measure on Y . Then ν|X is a complete finite Borel measure on X .

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. For homeomorphisms h : X → Y of separable metrizable spaces X and Y , if M ⊂ X , then FY (h[M ]) = h[FX (M )].