Associativity of the Morley Product of Invariant Measures in Nip Theories

Conant, Gabriel; Gannon, Kyle

doi:10.1017/jsl.2021.55

J. symb. log.

2021

DOI: 10.1017/jsl.2021.55

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Associativity of the Morley Product of Invariant Measures in Nip Theories

Gabriel Conant¹,

Kyle Gannon²

Abstract: In light of a gap found by Krupiński, we give a new proof of associativity for the Morley (or “nonforking”) product of invariant measures in NIP theories.

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Cited by 2 publications

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Definable convolution and idempotent Keisler measures

Chernikov

Gannon

2022

Isr. J. Math.

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We study idempotent measures and the structure of the convolution semigroups of measures over definable groups.We isolate the property of generic transitivity and demonstrate that it is sufficient (and necessary) to develop stable group theory localizing on a generically stable type, including invariant stratified ranks and connected components. We establish generic transitivity of generically stable idempotent types in important new cases, including abelian groups in arbitrary theories and arbitrary groups in rosy theories, and characterize them as generics of connected type-definable subgroups.Using tools from Keisler's randomization theory, we generalize some of these results from types to generically stable Keisler measures, and classify idempotent generically stable measures in abelian groups as (unique) translationinvariant measures on type-definable fsg subgroups. This provides a partial definable counterpart to the classical work of Rudin, Cohen and Pym for locally compact topological groups.Finally, we provide an explicit construction of a minimal left ideal in the convolution semigroup of measures for an arbitrary countable NIP group, from a minimal left ideal in the corresponding semigroup on types and a canonical measure constructed on its ideal subgroup. In order to achieve it, we in particular prove the revised Ellis group conjecture of Newelski for countable NIP groups.

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Definable convolution and idempotent Keisler measures

Chernikov

Gannon

2022

Isr. J. Math.

Self Cite

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Keisler measures in the wild

2023

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We investigate Keisler measures in arbitrary theories. Our initial focus is on Borel definability. We show that when working over countable parameter sets in countable theories, Borel definable measures are closed under Morley products and satisfy associativity. However, we also demonstrate failures of both properties over uncountable parameter sets. In particular, we show that the Morley product of Borel definable types need not be Borel definable (correcting an erroneous result from the literature). We then study various notions of generic stability for Keisler measures and generalize several results from the NIP setting to arbitrary theories. We also prove some positive results for the class of frequency interpretation measures in arbitrary theories, namely, that such measures are closed under convex combinations and commute with all Borel definable measures. Finally, we construct the first example of a complete type which is definable and finitely satisfiable in a small model, but not finitely approximated over any small model.

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Associativity of the Morley Product of Invariant Measures in Nip Theories

Abstract: In light of a gap found by Krupiński, we give a new proof of associativity for the Morley (or “nonforking”) product of invariant measures in NIP theories.

Cited by 2 publications

References 5 publications

Definable convolution and idempotent Keisler measures

Definable convolution and idempotent Keisler measures

Keisler measures in the wild

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