Super-rigid families of strongly Blackwell spaces

Abstract

We construct a complete subfield F F of P ( R ) P({\mathbf {R}}) , isomorphic to P ( R ) P({\mathbf {R}}) , of pairwise non-Borel-isomorphic rigid strong Blackwell subsets of R {\mathbf {R}} such that there are only ’very few’ measurable functions between any two members of F F . As a consequence, we obtain large chains and antichains of non-isomorphic rigid strong Blackwell subsets of R {\mathbf {R}} . Also, there is a collection of continuously many dense subsets of R {\mathbf {R}} such that any two of them differ only by two elements, but none of them is a continuous image of any other.