Palm pairs and the general mass-transport principle

TL;DR

This paper develops a general mass-transport principle for group actions on measure spaces, introducing new technical tools like Haar measure disintegration and characterizations of Palm pairs, applicable to non-unimodular groups.

Contribution

It provides a novel transport formula for Palm pairs and an intrinsic characterization of G-invariant measures, extending mass-transport principles to broader group actions.

Findings

Transport formula for Palm pairs established

Measurable disintegration of Haar measure achieved

General mass-transport principle derived for non-unimodular groups

Abstract

We consider a lcsc group G acting properly on a Borel space S and measurably on an underlying sigma-finite measure space. Our first main result is a transport formula connecting the Palm pairs of jointly stationary random measures on S. A key (and new) technical result is a measurable disintegration of the Haar measure on G along the orbits. The second main result is an intrinsic characterization of the Palm pairs of a G-invariant random measure. We then proceed with deriving a general version of the mass-transport principle for possibly non-transitive and non-unimodular group operations first in a deterministic and then in its full probabilistic form.