Measures and integrals in conditional set theory

Abstract

The aim of this article is to establish basic results in a conditional measure theory. The results are applied to prove that arbitrary kernels and conditional distributions are represented by measures in a conditional set theory. In particular, this extends the usual representation results for separable spaces.

21 pages, no figures, this is a final version accepted for publication in Set-Valued and Variational Analysis, compared with [v2] remark 5.5 added, presentation and description improved; after a major revision in [v2], where the general setting was simplified and extended applications to conditional distributions and disintegration were obtained (comprised in section 4)