Louveau's theorem for the descriptive set theory of internal sets

Abstract

AbstractWe give positive answers to two open questions from [15]. (1) For every set C countably determined over , if C is then it must be over , and (2) every Borel subset of the product of two internal sets X and Y all of whose vertical sections are can be represented as an intersection (union) of Borel sets with vertical sections of lower Borel rank. We in fact show that (2) is a consequence of the analogous result in the case when X is a measurable space and Y a complete separable metric space (Polish space) which was proved by A. Louveau and that (1) is equivalent to the property shared by the inverse standard part map in Polish spaces of preserving almost all levels of the Borel hierarchy.