On Maximal Ranges of Vector Measures for Subsets and Purification of Transition Probabilities

TL;DR

This paper explores the maximal ranges of vector measures in measurable spaces, showing their existence in two dimensions and using this to improve a classical purification theorem.

Contribution

It establishes the existence of maximal ranges for 2D vector measures and applies this to strengthen the Dvoretzky-Wald-Wolfowitz purification theorem.

Findings

Maximal ranges exist for 2D vector measures.

Maximal ranges do not necessarily exist in higher dimensions.

Strengthened purification theorem for two measures.

Abstract

Consider a measurable space with an atomless finite vector measure. This measure defines a mapping of the $\sigma$-field into an Euclidean space. According to the Lyapunov convexity theorem, the range of this mapping is a convex compactum. Similar ranges are also defined for measurable subsets of the space. Two subsets with the same vector measure may have different ranges. We investigate the question whether, among all the subsets having the same given vector measure, there always exists a set with the maximal range of the vector measure. The answer to this question is positive for two-dimensional vector measures and negative for higher dimensions. We use the existence of maximal ranges to strengthen the Dvoretzky-Wald-Wolfowitz purification theorem for the case of two measures.