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Geometric properties of disintegration of measures

Renata Possobon, Christian S. Rodrigues

TL;DR

This work embeds disintegration of measures into the geometric framework of Optimal Transport to obtain a fibre-wise, transport-based perspective. It establishes a disintegration theorem that yields a measurable family of conditional measures $\{\mu_y\}$ with $\mu=\nu\otimes\mu_y$, and introduces disintegration maps $f$ that send base points to conditional measures in $\mathscr{P}(X)$ under $W_p$-metrics. By defining transport classes and linking to Monge–Kantorovich theory, the paper reframes MK problems in terms of fiberwise decompositions and displacement interpolations. It then analyzes when disintegration maps are (nearly) weakly continuous, proves rigidity results for absolute continuity along measure-paths, and highlights cases where the map is an isometry under metric-measure foliations. The results provide a robust geometric toolkit for studying regularity (continuity and absolute continuity) of conditional measures and their paths, with implications for ergodic theory, foliations, and dynamical systems.

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.

Geometric properties of disintegration of measures

TL;DR

This work embeds disintegration of measures into the geometric framework of Optimal Transport to obtain a fibre-wise, transport-based perspective. It establishes a disintegration theorem that yields a measurable family of conditional measures with , and introduces disintegration maps that send base points to conditional measures in under -metrics. By defining transport classes and linking to Monge–Kantorovich theory, the paper reframes MK problems in terms of fiberwise decompositions and displacement interpolations. It then analyzes when disintegration maps are (nearly) weakly continuous, proves rigidity results for absolute continuity along measure-paths, and highlights cases where the map is an isometry under metric-measure foliations. The results provide a robust geometric toolkit for studying regularity (continuity and absolute continuity) of conditional measures and their paths, with implications for ergodic theory, foliations, and dynamical systems.

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.
Paper Structure (5 sections, 15 theorems, 88 equations, 5 figures)

This paper contains 5 sections, 15 theorems, 88 equations, 5 figures.

Table of Contents

  1. Introduction
  2. Spaces of probability measures, Wasserstein spaces and Optimal Transport
  3. Disintegration of measures
  4. Disintegration maps
  5. Absolute continuity from disintegration maps

Key Result

Theorem 2.4

Fed69. Let $M$ be a locally compact metric space and $N$ a separable metric space. Consider $\mu$ a Borel measure on $M$, $A \subset M$ a measurable set with finite measure and $f : M \to N$ a measurable map. Then, for each $\delta > 0$ there is a closed set $K \subset A$, with $\mu(A \backslash K)

Figures (5)

  • Figure 1: Representation of $\mathcal{F}^{s} = \{\{x\} \times D^{2} \}_{x \in S^{1}}$.
  • Figure 2: Transport plan 1.
  • Figure 3: Transport plan 2.
  • Figure 4: Transport plan 3.
  • Figure 5: Idea of a metric measure foliation as a space fibration.

Theorems & Definitions (43)

  • Definition 2.1
  • Definition 2.2: Wasserstein distance
  • Definition 2.3: Wasserstein space
  • Theorem 2.4
  • Lemma 2.5: Gluing Lemma
  • Theorem A
  • proof
  • Corollary 3.1
  • Example 3.2
  • Example 3.3
  • ...and 33 more