# A unifying view on some problems in probability and statistics

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## 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.

## Keywords

Compatibility of conditional distributions De Finetti’s coherence principle Equivalent martingale measure Equivalent probability measure with given marginals Finitely additive probability Fundamental theorem of asset pricing## References

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