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.
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