Borel density, the marginal problem and isomorphism types of analytic sets

Document Type

Article

Publication Date

1-1-1984

Abstract

The present work arises out of an ongoing study of the existence of probabilities with prescribed marginals, in particular, an attempt to determine exactly for which spaces property (V) and the Kolmogoroff consistency theorem hold. To this end, we have introduced the concept of Borel-density, in fact an infinite hierarchy of Borel-densities (see Proposition 3). Their relationship to the marginal problem is explored in Propositions 9 and 10: density of order 3 implies property (V) and, in the presence of order 2 density, is equivalent with it. Propositions 11 and 12 treat Kolmogoroff consistency problems: infinite-order Borel-density is sufficient for Kolmogoroff’s theorem to hold; as a consequence, there are highly non-measurable spaces over which the theorem obtains. Finally, and perhaps most intriguingly, there are applications of these results to the (open) problem of determining the isomorphism types of analytic sets. Proposition 13 asserts that if X1 and X2 are uncountable separable spaces such that X1 × X2 is Borel-dense, then Xx and X2 are standard. This last result improves a theorem of R. D. Mauldin (1976) to the effect that if an analytic (non-Borel) subset A of the unit interval has totally imperfect complement, then A is not isomorphic with An, n ≥ 2. A consequence of our Proposition 13 (Corollary 7) is that such an A is not isomorphic with any product B × C of uncountable spaces B and C. We do not use the method of Lusin sieves. The definition of wth order Borel-density bears some formal resemblance to certain work of Cox (1980) on Lusin properties for Cartesian products, but the exact link seems unclear. © 1984 by Pacific Journal of Mathematics.

Publication Title

Pacific Journal of Mathematics

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