Keisler Measure

A Keisler measure is a finitely additive probability measure on the Boolean algebra of sets definable by first-order formulas in a fixed tuple of variables and with parameters from a specified structure. Thus it assigns values in , assigns 1 to the whole domain, and adds the measures of disjoint definable sets.

A complete type gives a zero-one valued example: a formula has measure 1 precisely when it belongs to the type. More generally, a finite convex combination of such examples is a Keisler measure. This extends the study of types to a probabilistic setting.

Conant et al. (2026) proved that three proposed notions of generic stability coincide for Borel-definable global Keisler measures, in both discrete and continuous logic. The conditions are the frequency interpretation property, definability together with generic stability of the canonical random extension, and self-averaging. These compare uniform approximation by finite samples, stability after passing to a randomized theory, and convergence of averages along the model-theoretic analogue of independent samples.

The new reverse implications were first obtained with ChatGPT 5.5, independently reproduced using Kimi K3 and Claude Fable 5, and reorganized and checked by the authors. Independent external review had not been reported as of Sep. 7, 2026.