  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.
