A canonical hidden-variable space

Abstract

The hidden-variable question is whether or not various properties — randomness or correlation, for example — that are observed in the outcomes of an experiment can be explained via introduction of extra (hidden) variables which are unobserved by the experimenter. The question can be asked in both the classical and quantum domains. In the latter, it is fundamental to the interpretation of the quantum formalism (Bell [1], Kochen and Specker [9], and others). In building a suitable mathematical model of an experiment, the physical set-up will guide us on how to model the observable variables — i.e., the measurement and outcome spaces. But, by definition, we cannot know what structure to put on the hidden-variable space. Nevertheless, we show that, under a measure-theoretic condition, the hidden-variable question can be put into a canonical form. The condition is that the σ-algebras on the measurement and outcome spaces are countably generated. An argument using a classical result on isomorphisms of measure algebras then shows that the hidden-variable space can always be taken to be the unit interval equipped with the Lebesgue measure on the Borel sets.