Geometric properties of disintegration of measures

TL;DR

This paper explores the geometric aspects of measure disintegration, establishing a disintegration theorem linked to optimal transport, and analyzing the regularity and absolute continuity of disintegrated measures.

Contribution

It introduces a new disintegration theorem connected to optimal transport and provides conditions for the regularity and absolute continuity of disintegrated measures.

Findings

Disintegration maps can be weakly continuous under certain conditions.

Conditions for disintegration into absolutely continuous measures are identified.

A rigidity condition for absolute continuity in measure disintegration is established.

Abstract

In this paper, we study a connection between disintegration of measures and geometric properties of probability spaces. We prove a disintegration theorem, addressing disintegration from the perspective of an optimal transport problem. We look at the disintegration of transport plans, which are used to define and study disintegration maps. Using these objects, we study the regularity and absolute continuity of disintegration of measures. In particular, we exhibit conditions for which the disintegration map is weakly continuous and one can obtain a path of measures given by this map. We show a rigidity condition for the disintegration of measures to be given into absolutely continuous measures.