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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 X1 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 n-th 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.
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