Measures and Stability in a Model, revisited

This article is written in honor 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 We expand on an unpublished research note of the second author.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) local 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 conclude with observations regarding the NOP and local Keisler measures in the (properly) stable context. Finally, we provide two proofs that the evaluation map on pairs of local Keisler measures is stable (in continuous logic). The first follows almost immediately from the work of Ben Yaacov and Keisler on the randomization; the other proof follows from the VC theorem.