Transference of measures via disintegration

TL;DR

This paper investigates the structure of positive measures on product spaces representing functionals on vector-valued continuous functions, using disintegration techniques to enhance measure transference results and reveal an order structure.

Contribution

It introduces an alternative disintegration-based approach to measure transference, strengthening previous results and uncovering a rich order structure on these measures.

Findings

Strengthened measure transference results.

Identified maximal and minimal measures.

Established relation to Choquet order.

Abstract

Given a compact space $K$ and a Banach space $E$ we study the structure of positive measures on the product space $K\times B_{E^*}$ representing functionals on $C(K,E)$, the space of $E$-valued continuous functions on $K$. Using the technique of disintegration we provide an alternative approach to the procedure of transference of measures introduced by Batty (1990). This enables us to substantially strengthen some of his results, to discover a rich order structure on these measures, to identify maximal and minimal elements and to relate them to the classical Choquet order.