Smooth measures and the canonical retraction in NIP theories

TL;DR

This work extends Simon's canonical retraction $F_M$ from $M$-invariant types to Keisler measures in NIP theories, establishing that the same core properties hold for measures. It introduces smooth measures and an amalgam framework, develops an alternative pair-model construction of the canonical retraction that coincides with the standard one, and proves a definability–commutation equivalence: an $M$-invariant measure $\mu$ is definable over $M$ iff it commutes with $F_M(\mu)$. The results include that $F_M$ is an affine, continuous retraction on the space of $M$-invariant measures, with $F_M(\mu)=\mu$ for measures finitely satisfiable in $M$, and that $F_M(\nu\otimes\mu)=\nu\otimes F_M(\mu)$ when $\nu$ is finitely satisfiable in $M$. Collectively, these contributions provide a robust bridge between type- and measure-theoretic canonical retractions, and extend the toolkit for analyzing definable and generically stable measures in pair-model contexts within NIP theories.

Abstract

We show that the results proved by Simon on the canonical retraction $F_M$ from the space of $M$-invariant types onto the space of types finitely satisfiable in $M$ remain true over measures. We also make another construction of the canonical retraction for measures, mimicking what Simon did for types, and show that it coincides with Simon's canonical retraction for measures. To do so, we make extensive use of smooth measures.

Theorems & Definitions (56)

• Theorem 2
• Definition 1.2
• Remark
• Remark
• Definition 2.1
• Claim
• proof
• Lemma 2.2
• proof
• Proposition 2.3