Blackwell spaces and generalized Lusin sets (Q1068226)

Let \({\mathcal B}\) be the Borel \(\sigma\)-algebra of a separable metric space X. Say that X has the Blackwell property if whenever \({\mathcal C}\) is a countably generated (c.g.) sub-\(\sigma\)-algebra of \({\mathcal B}\) separating points of X, then \({\mathcal C}={\mathcal B}\). Say that X has the strong Blackwell property if whenever \({\mathcal C}\) and \({\mathcal D}\) are c.g. sub- \(\sigma\)-algebras of \({\mathcal B}\) with the same atoms, then \({\mathcal C}={\mathcal D}.\) In previous work, the first author proved that if \(X\subseteq {\mathbb{R}}\) has totally imperfect complement (\({\mathbb{R}}\setminus X\) contains no Cantor-like set), then X has the Blackwell property if and only if X has the strong Blackwell property. Also, if \(X\subseteq Y\subseteq {\mathbb{R}},\) then Y has the strong Blackwell property. These results are generalized. In particular, it is shown that if \(A\subseteq {\mathbb{R}}\) is an uncountable set with the Blackwell property and if \(Y\cap B\) is countable for all Borel \(B\subseteq {\mathbb{R}}\setminus A,\) then \(A\cup Y\) has the Blackwell property. A consequence of this is a recent result of Jasinski: if \(A\subseteq {\mathbb{R}}\) is an uncountable analytic set, and if \(Y\cap C\) is countable for each constituent of \({\mathbb{R}}\setminus A\), then \(A\cup Y\) has the Blackwell property.