Approximate equivalence relations

Abstract

We study approximate equivalence relations up to commensurability, in the presence of a definable measure. As a basic framework, we give a presentation of probability logic based on continuous logic. Hoover's normal form is valid here; if one begins with a discrete logic structure, it reduces arbitrary formulas of probability logic to correlations between quantifier-free formulas. We completely classify binary correlations in terms of the Kim-Pillay space, leading to strong results on the interpretative power of pure probability logic over a binary language. Assuming higher amalgamation of independent types, we prove a higher stationarity statement for such correlations. We also give a short model-theoretic proof of a categoricity theorem for continuous logic structures with a measure of full support, generalizing theorems of Gromov-Vershik and Keisler, and often providing a canonical model for a complete pure probability logic theory. These results also apply to local probability logic, providing in particular a canonical model for a local pure probability logic theory with a unique 1-type and geodesic metric. For sequences of approximate equivalence relations with an 'approximately unique' probability logic 1-type, we obtain a structure theorem generalizing the `Lie model' theorem for approximate subgroups. The models here are Riemannian homogeneous spaces, fibered over a locally finite graph. Specializing to definable graphs over finite fields, we show that after a finite partition, a definable binary relation converges in finitely many self-compositions to an equivalence relation of geometric origin. For NIP theories, pursuing a question of Pillay's, we prove an archimedean finite-dimensionality statement for the automorphism groups of definable measures, acting on a given type of definable sets.