Ordered Probability Spaces

TL;DR

This paper establishes conditions under which probability measures ordered by a cone-induced stochastic order can be approximated by finitely supported measures, enabling the extension of operator mean inequalities to measure spaces.

Contribution

It provides a method to approximate measures in a cone with finite support measures preserving order, facilitating the extension of inequalities to measure spaces on cones.

Findings

Order approximation of measures in cone spaces

Extension of operator mean inequalities to measure spaces

Monotonicity of the Karcher geometric mean on measure spaces

Abstract

Let $C$ be an open cone in a Banach space equipped with the Thompson metric with closure a normal cone. The main result gives sufficient conditions for Borel probability measures $\mu,\nu$ on $C$ with finite first moment for which $\mu\leq \nu$ in the stochastic order induced by the cone to be order approximated by sequences $\{\mu_n\},\{\nu_n\}$ of uniform finitely supported measures in the sense that $\mu_n\leq \nu_n$ for each $n$ and $\mu_n\to \mu$, $\nu_n\to \nu$ in the Wasserstein metric. This result is the crucial tool in developing a pathway for extending various inequalities on operator and matrix means, which include the harmonic, geometric, and arithmetic operator means on the cone of positive elements of a $C^*$-algebra, to the space $\mathcal{P}^1(C)$ of Borel measures of finite first moment on $C$. As an illustrative particular application, we obtain the monotonicity of the Karcher geometric mean on $\mathcal{P}^1(\mathbb{A}^+)$ for the positive cone $\mathbb{A}^+$ of a $C^*$-algebra $\mathbb{A}$.