# Measures and integrals in conditional set theory

## TL;DR

This paper develops foundational results in conditional measure theory, demonstrating how kernels and distributions can be represented by measures within a conditional set framework, extending classical representation theorems.

## Contribution

It introduces basic results in conditional measure theory and extends representation theorems to non-separable spaces within this framework.

## Key findings

- Established foundational results in conditional measure theory.
- Proved that kernels and distributions are representable by measures in conditional set theory.
- Extended classical representation results to broader spaces.

## 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.