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On Isomorphic Probability Spaces

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Part of the Lecture Notes in Mathematics book series (SEMPROBAB,volume 2006)

Abstract

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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Reference

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

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Correspondence to Claude Dellacherie .

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© 2011 Springer-Verlag Berlin Heidelberg

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Dellacherie, C. (2011). On Isomorphic Probability Spaces. In: Donati-Martin, C., Lejay, A., Rouault, A. (eds) Séminaire de Probabilités XLIII. Lecture Notes in Mathematics(), vol 2006. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-15217-7_6

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