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• W1979882324 abstract "Under the assumption of CH (continuum hypothesis) we produce a strongly Blackwell set whose product with a standard space and whose intersection with an analytic set are not Blackwell. Previously, such examples were known to exist only under Martin's Axiom (MA) and not-CH. 0. Introduction. Blackwell sets, which function in the category of measurable spaces as an analogon to compact spaces in the topological category, have been the focus of some recent papers [1-5, 8]. For the motives behind such study, as well as a survey of basic results, the reader is referred to [1]. In particular, Jakub Jasinski [5] has shown that, with the assumption of Martin's Axiom (MA) and the negation of the Continuum Hypothesis (CH), these spaces are poorly behaved under the operations of product and intersection. By making use of some theory developed with the assistance of Bhaskara Rao [8], it is possible to demostrate similar pathologies under the assumption of CH. A construction is given in the proposition of ?3 infra. 1. Preliminaries. A measurable space (X, B) is separable if its Borel structure B = B (X) is countably generated (c.g.) and contains all singleton sets drawn from X. If Y C X, then (Y, B(Y)) is again separable, where B (Y) = {BnY: B e B(X)}. A aeparable space (S, B) is standard if there is a complete separable metrizable (i.e., Polish) topology on S whose Borel a-field is B = B(S). A subset of a separable spaces is analytic if it is a measurable image of a standard space. Any two uncountable standard spaces are Borel-isomorphic [6, pp. 487-489]. A separable space X has the Blackwell property if B (X) does not properly contain any c.g. sub-a-algebra separating points of X. A separable space X has the strong Blackwell property if any two c.g. sub-u-algebras of B (X) with the same atoms coincide. Every analytic space has the strong Blackwell property. For proof of this and other facts concerning Blackwell sets, consult the monograph [1]. Let S be a standard space. A subclass I of B(S) is a u-ideal if it is closed under countable unions and has the property that N n B E I whenever N E I and B e B(S). If N C S, we write (N) to denote the subset (N) = (N x S) U (S x N) of S x S. A subset R of S x S is I-reticulate if R c (N) for some N e I. A subset X of S is I-dense in S if X has nonvoid intersection with every set in B(S)I. A set X C S is I-dense of order 2 in S if X x X meets every set R e B(S x S) which Received by the editors September 20, 1986. 1980 Mathematics Subject Classification (1985 Revision). Primary 54H05; Secondary 28A05, 03E15." @default.
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• W1979882324 date "1987-04-01" @default.
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• W1979882324 title "Combinatorial properties for Blackwell sets" @default.
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• W1979882324 doi "https://doi.org/10.1090/s0002-9939-1987-0911043-7" @default.
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