On Isomorphic Probability Spaces

  • Claude DellacherieEmail author
Part of the Lecture Notes in Mathematics book series (LNM, volume 2006)


In the appendix to his contribution (Laurent, On standardness and I-cosiness, this volume) to this volume, Stéphane Laurent recalls that if a probability space \((\Omega,\mathcal{A}, \mathbb{P})\) is embedded in another probability space \((\Omega ',\mathcal{A}', \mathbb{P}')\), to every r.v. X on Ω the embedding associates a r.v. X′ on Ω′. More precisely, his Lemma 5.5 states this property when X is valued in a Polish space E. Michel Émery has asked me the following question: is completeness of E really needed, or does the property more generally hold for separable, non complete metric spaces? By means of a counter-example, this short note shows that completeness cannot be dispensed of.

Isomorphic probability space Counterexample 


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  1. 1.
    Sminaire de Probabilits XLIII, Lecture Notes in Math., vol. 2006, Springer, New York (2011)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2011

Authors and Affiliations

  1. 1.Laboratoire Raphaël SalemC.N.R.S. et Université de Rouen Avenue de l’UniversitéSaint-Étienne-du-RouvrayFrance

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