On Isomorphic Probability Spaces

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.