By Togo Nishiura

ISBN-10: 0521875560

ISBN-13: 9780521875561

Absolute measurable house and absolute null house are very outdated topological notions, constructed from recognized proof of descriptive set idea, topology, Borel degree concept and research.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 likelihood measures at the Cantor house, are one of many issues 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 idea, Hausdorff degree and Hausdorff size, and geometric degree concept.

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**Sample text**

There is no loss in assuming that Y is a subspace of the Hilbert cube [0, 1] N . Let X be a subset of Y that is well ordered by the relation R and denote its indexing by { xα : α < η }. We shall assume that it is not an absolute null space contained in Y and derive a contradiction. There is a continuous, complete Borel measure µ on [0, 1] N such that 1 = µ [0, 1] N ≥ µ∗ (X ) > 0. As X is well ordered, there is an η0 such that µ { xα : α < η0 } > 0, and such that µ { xα : α < β } = 0 whenever β < η0 .

A topology τ on the set S corresponding to this separable σ -algebra A results in a topological embedding f of S into {0, 1}N . 4. Grzegorek’s cardinal number κG 21 ν = f# µ is a nontrivial, continuous, complete, finite Borel measure on {0, 1}N and ν ∗ f [S] = µ(S) > 0. As card( f [S]) = card(S), we have κ0 ≤ card(S). Therefore, κ0 ≤ κG and the lemma is proved. ✷ We now give Grzegorek’s theorem. 44 (Grzegorek). Let S ⊂ {0, 1}N and let µ be a continuous, complete, finite Borel measure on {0, 1}N such that card(S) = κ0 and µ∗ (S) > 0.

1) X contains an uncountable absolute null space. (2) X contains an uncountable absolute measurable space. (3) X has a transfinite sequence Bα , α < ω1 , that is m-convergent in X . Proof. As absolute null spaces are absolute measurable spaces, we have that statement (1) implies statement (2). To prove that statement (2) implies statement (3), suppose that X is an absolute measurable space that is contained in X . If X is already an uncountable absolute null space, then X contains a subset X0 with card(X0 ) = ℵ1 .

### Absolute Measurable Spaces (Encyclopedia of Mathematics and its Applications) by Togo Nishiura

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