On NIP and invariant measures

2007/10/11 by Ehud Hrushovski, Anand Pillay, Hrushovski, Ehud +1 · 4 citations
Computer Science · Mathematics · #03C45 #03C60 #20F67 #Advanced Operator Algebra Research #Advanced Topology and Set Theory #Computability, Logic, AI Algorithms #FOS: Mathematics #Logic (math.LO)

paper · pdf · doi:10.48550/arxiv.0710.2330

openalex publication_date 2007/10/11 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We study forking, Lascar strong types, Keisler measures and definable groups, under an assumption of NIP (not the independence property), continuing aspects of math.LO/0607442. Among key results are: (i) if p = tp(b/A) does not fork over A then the Lascar strong type of b over A coincides with the compact strong type of b over A and any global nonforking extension of p is Borel definable over bdd(A) (ii) analogous statements for Keisler measures and definable groups, including the fact that G⁰⁰⁰ = G⁰⁰ for G definably amenable, (iii) definitions, characterizations and properties of "generically stable" types and groups (iv) uniqueness of translation invariant Keisler measures on groups with finitely satisfiable generics (vi) A proof of the compact domination conjecture for definably compact commutative groups in o-minimal expansions of real closed fields.

Cited by

• Upside down and backwards
• On idempotent measure conjecture and decomposition of invariant measures
• Generic sampling and invariant measures on the space of k-uniform hypergraphs
• The number of countable models of first-order theories

Related

• Groups, measures, and the NIP
• Invariant measures in simple and in small theories
• On f-generic types in NIP groups
• On idempotent measure conjecture and decomposition of invariant measures
• Definable convolution and idempotent Keisler measures III. Generic stability, generic transitivity, and revised Newelski's conjecture