Sure Wins, Separating Probabilities and the Representation of Linear Functionals

TL;DR

This paper explores conditions for convex cones to admit probability measures with bounded expectations and characterizes linear functionals representable as finitely additive expectations, providing new insights into positive functionals and Riesz decomposition.

Contribution

It introduces novel conditions for the existence of probability measures on convex cones and characterizes linear functionals as finitely additive expectations, extending classical results.

Findings

Characterization of convex cones admitting probability measures

Representation of linear functionals as finitely additive expectations

A version of Riesz decomposition based on these properties

Abstract

We discuss conditions under which a convex cone $\K\subset \R^{\Omega}$ admits a probability $m$ such that $\sup_{k\in \K} m(k)\leq0$. Based on these, we also characterize linear functionals that admit the representation as finitely additive expectations. A version of Riesz decomposition based on this property is obtained as well as a characterisation of positive functionals on the space of integrable functions