Fubini-Tonelli type theorem for non product measures in a product space

TL;DR

This paper extends Fubini-Tonelli type theorems to non-product measures in product spaces, establishing conditions for iterated integrals and providing an application in Optimal Transport.

Contribution

It introduces a new theorem for iterated integrals with non-product measures and constructs a family of measures on the second space indexed by the first, outside a negligible set.

Findings

Existence of a measure family indexed by the first space points

Representation of the original measure via these measures

Application to Optimal Transport problem

Abstract

I prove a theorem about iterated integrals for non-product measures in a product space. The first task is to show the existence of a family of measures on the second space, indexed by the points on of the first space (outside a negligible set), such that integrating the measures on the index against the first marginal gives back the original measure (see Theorem 2.1). At the end, I give a simple application in Optimal Transport.