By Togo Nishiura

Absolute measurable area and absolute null house are very previous topological notions, constructed from famous evidence of descriptive set thought, topology, Borel degree idea 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 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. life 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 one of several 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 concept, Hausdorff degree and Hausdorff size, and geometric degree conception.

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**Example text**

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]. 37. Let X be a set with card(X ) = ℵ1 , and let K be a class of subsets of X with the following properties: (1) K is a σ -ideal, (2) the union of K is X , (3) K has a subclass G with card(G) = ℵ1 and the property that each member of K is contained in some member of G, (4) the complement of each member of K contains a set with cardinality ℵ1 that belongs to K.

We turn to the converse next. 1. Existence of positive measures. It is not immediate that positive measures exist if FX (X ) = ∅. 14. For separable metrizable spaces X , MEASpos (X ) is not empty if and only if FX (X ) is not empty. Proof. We shall use F for FX in the proof. Suppose MEASpos (X ) is not empty. Then there exists a measure µ such that support(µ) = F(X ) = ∅. Suppose that F(X ) = ∅. Note that µ X \F(X ) = 0 for every continuous, complete, finite Borel measure µ on X . Let U0 , U1 , .

There exists a subspace X of {0, 1}N such that X is an absolute null space with card(X ) = κG . 11 A cardinal number κ will be identified with the minimal initial segment of ordinal numbers whose cardinal number is κ. 21 we have the following corollary, where P(X ) is the power set of X . 46. If (X , µ, P(X )) is a nonnegative real-valued, continuous measure space with card(X ) = κG , then µ is identically equal to 0. 3. abNULL is not preserved by Borel measurable maps. Let us begin with a simple proposition.