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Ordered Probability Spaces

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 $μ,ν$ on $C$ with finite first moment for which $μ\leq ν$ in the stochastic order induced by the cone to be order approximated by sequences $\{μ_n\},\{ν_n\}$ of uniform finitely supported measures in the sense that $μ_n\leq ν_n$ for each $n$ and $μ_n\to μ$, $ν_n\to ν$ 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 $\mathbb{A}$.