A unifying view on some problems in probability and statistics

Abstract

Let $L$ be a linear space of real random variables on the measurable space $(\varOmega ,\mathcal {A})$. Conditions for the existence of a probability $P$ on $\mathcal {A}$ such that $E_P|X|<\infty$ and $E_P(X)=0$ for all $X\in L$ are provided. Such a $P$ may be finitely additive or $\sigma$-additive, depending on the problem at hand, and may also be requested to satisfy $P\sim P_0$ or $P\ll P_0$ where $P_0$ is a reference measure. As a motivation, we note that a plenty of significant issues reduce to the existence of a probability $P$ as above. Among them, we mention de Finetti’s coherence principle, equivalent martingale measures, equivalent measures with given marginals, stationary and reversible Markov chains, and compatibility of conditional distributions.