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Reconciliation of probability measures

Part of: Foundations of probability theory

Published online by Cambridge University Press:  17 December 2018

JEAN BÉRARD  and
NICOLAS JUILLET
JEAN BÉRARD
Affiliation:
Institut de Recherche Mathématique Avancée, UMR 7501, Université de Strasbourg et CNRS, 7 rue René Descartes, 67 000 Strasbourg, France emails: jean.berard@math.unistra.fr; nicolas.juillet@math.unistra.fr
NICOLAS JUILLET*
Affiliation:
Institut de Recherche Mathématique Avancée, UMR 7501, Université de Strasbourg et CNRS, 7 rue René Descartes, 67 000 Strasbourg, France emails: jean.berard@math.unistra.fr; nicolas.juillet@math.unistra.fr
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Abstract

We discuss the reconciliation problem between probability measures: given n⩾2 probability spaces $(\Omega,{\mathcal{F}}_1,{\mathbb{P}}_1),\ldots,(\Omega,{\mathcal{F}}_n,{\mathbb{P}}_n)$ with a common sample space, does there exist an overall probability measure ${\mathbb{P}} \ \text{on} \ {\mathcal{F}} = \sigma({\mathcal{F}}_1,\ldots,{\mathcal{F}}_n)$ such that, for all i, the restriction of ${\mathbb{P}} \ \text{to} \ {\mathcal{F}}_i$ coincides with ${\mathbb{P}}_i$? General criteria for the existence of a reconciliation are stated, along with some counterexamples that highlight some delicate issues. Connections to earlier (recent and far less recent) work are discussed, and elementary self-contained proofs for the various results are given.

Keywords

Optimal transportjoint lawsigma-fieldsFarkas’ LemmaStrassen theorem

MSC classification

Primary: 60A05: Axioms; other general questions
Secondary: 90C08: Special problems of linear programming (transportation, multi-index, etc.)
Type
Papers
Information
European Journal of Applied Mathematics , Volume 30 , Special Issue 6: Applied Optimal Transport , December 2019 , pp. 1300 - 1310
Copyright
© Cambridge University Press 2018 

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