Model theory of probability spaces

This expository paper treats the model theory of probability spaces using the framework of continuous $[0,1]$-valued first order logic. The metric structures discussed, which we call probability algebras, are obtained from probability spaces by identifying two measurable sets if they differ by a set of measure zero. The class of probability algebras is axiomatizable in continuous first order logic; we denote its theory by $Pr$. We show that the existentially closed structures in this class are exactly the ones in which the underlying probability space is atomless. This subclass is also axiomatizable; its theory $APA$ is the model companion of $Pr$. We show that $APA$ is separably categorical (hence complete), has quantifier elimination, is $\omega$-stable, and has built-in canonical bases, and we give a natural characterization of its independence relation. For general probability algebras, we prove that the set of atoms (enlarged by adding $0$) is a definable set, uniformly in models of $Pr$. We use this fact as a basis for giving a complete treatment of the model theory of arbitrary probability spaces. The core of this paper is an extensive presentation of the main model theoretic properties of $APA$. We discuss Maharam's structure theorem for probability algebras, and indicate the close connections between the ideas behind it and model theory. We show how probabilistic entropy provides a rank connected to model theoretic forking in probability algebras. In the final section we mention some open problems.