A separation theorem for Σ¹₁ sets

Abstract

In this paper, we show that the notion of Borel class is, roughly speaking, an effective notion. We prove that if a set A is both ∏ ξ 0 \prod _\xi ^0 and Δ 1 1 \Delta _1^1 , it possesses a Π ξ 0 \Pi _\xi ^0 -code which is also Δ 1 1 \Delta _1^1 . As a by-product of the induction used to prove this result, we also obtain a separation result for Σ 1 1 \Sigma _1^1 sets: If two Σ 1 1 \Sigma _1^1 sets can be separated by a Π ξ 0 \Pi _\xi ^0 set, they can also be separated by a set which is both Δ 1 1 \Delta _1^1 and Π ξ 0 \Pi _\xi ^0 . Applications of these results include a study of the effective theory of Borel classes, containing separation and reduction principles, and an effective analog of the Lebesgue-Hausdorff theorem on analytically representable functions. We also give applications to the study of Borel sets and functions with sections of fixed Borel class in product spaces, including a result on the conservation of the Borel class under integration.