  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.

Logic