Probability spaces become metric structures once measurable sets that differ by a null set are identified, and the resulting probability algebras are exactly the models of an axiomatizable continuous first-order theory Pr. Students learn why the existentially closed models are the atomless ones, why their theory APA is the model companion of Pr and is complete, separably categorical, quantifier-eliminable and omega-stable, and how the definability of the set of atoms reduces every probability algebra to its atomless part. The treatment also connects model-theoretic independence to probabilistic conditional independence, canonical bases to conditional expectation, Maharam's structure theorem to saturation, and forking to probabilistic entropy.
This expository paper by Alexander Berenstein and C. Ward Henson examines axiomatizable classes within continuous first-order logic, demonstrating that the class of probability algebras is axiomatiza…