Measures and stability in a model, revisited

TL;DR

This paper investigates local Keisler measures in stable formulas, showing they are weighted sums of types, and provides elementary and functional analytic proofs of measure properties, with implications for stability theory.

Contribution

It establishes that local Keisler measures in stable formulas are countable sums of types and offers new proofs for measure properties in this context.

Findings

Local Keisler measures are weighted sums of types.

Elementary proof of Morley product commutativity.

Double limit property extends to local measures.

Abstract

This article is written in celebration of the 8th Kazakh-French Logical Colloquium. We expand on an unpublished research note of the second author. We record some results concerning local Keisler measures with respect to a formula which is stable in a model. We prove that in this context, every local Keisler measure on the associated local type space is a weighted sum of (at most countably many) types. Using this observation, we give an elementary proof of the commutativity of the Morley product in this context. We then give a functional analytic proof that the double limit property lifts to the appropriate evaluation map on pairs of local measures. We end with some comments on the NOP and local measures in the (properly) stable context.