Submission date: 3 August 2023

Abstract:

We prove a number of results relating the concepts of Keisler measures, generic stability, randomizations, and NIP formulas. Among other things, we do the following: (1) We introduce the notion of a Keisler-Morley measure, which plays the role of a Morley sequence for a Keisler measure. We prove that if μ is fim over M, then for any Keisler-Morley measure λ in μ over M and any formula φ(x,b), lim_{i → ∞}λ(φ(x_i,b)) = μ(φ(x,b)). We also show that any measure satisfying this conclusion must be fam. (2) We study the map, defined by Ben Yaacov, taking a definable measure μ to a type r_μ in the randomization. We prove that this map commutes with Morley products, and that if μ is fim then r_μ is generically stable. (3) We characterize when generically stable types are closed under Morley products by means of a variation of ict-patterns. Moreover, we show that NTP_2 theories satisfy this property. (4) We prove that if a local measure admits a suitably tame global extension, then it has finite packing numbers with respect to any definable family. We also characterize NIP formulas via the existence of tame extensions for local measures.

Mathematics Subject Classification:

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Full text arXiv 2308.01801: pdf, ps.