• Corpus ID: 232240246

Keisler measures in the wild

@inproceedings{Conant2021KeislerMI,
  title={Keisler measures in the wild},
  author={Gabriel Conant and Kyle Gannon and James Hanson},
  year={2021}
}
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… 
Sequential approximations for types and Keisler measures
TLDR
It is shown that both generically stable types and Keisler measures which are finitely satisfiable over a countable model (in NIP theories) are sequentially approximated.
ASSOCIATIVITY OF THE MORLEY PRODUCT OF INVARIANT MEASURES IN NIP THEORIES
TLDR
A new proof of associativity for the Morley (or “nonforking”) product of invariant measures in NIP theories is given.
Generic Stability and Modes of Convergence
We study generically stable types/measures in both classical and continuous logics, and their connection with randomization and modes of convergence of types/measures.
Definable convolution and idempotent Keisler measures II
We study convolution semigroups of invariant/finitely satisfiable Keisler measures in NIP groups. We show that the ideal (Ellis) subgroups are always trivial and describe minimal left ideals in the
Dependent measures in independent theories
We introduce the notion of dependence, as a property of a Keisler measure, and generalize several results of [HPS13] on generically stable measures (in NIP theories) to arbitrary theories. Among

References

SHOWING 1-10 OF 36 REFERENCES
On NIP and invariant measures
We study forking, Lascar strong types, Keisler measures and definable groups, under an assumption of NIP (not the independence property), continuing aspects of the paper [13]. Among key results are
Groups, measures, and the NIP
We discuss measures, invariant measures on definable groups, and genericity, often in an NIP (failure of the independence property) environment. We complete the proof of the third author’s
Descriptive Set Theory and Forcing: How to Prove Theorems about Borel Sets the Hard Way
TLDR
These are lecture notes from a course I gave at the University of Wisconsin during the Spring semester of 1993 about Borel hierarchies and Louveau's Theorem on $\Pi^0_\alpha$ hyp-sets has a simpler proof using forcing.
Generically stable and smooth measures in NIP theories
We formulate the measure analogue of generically stable types in first order theories with NIP (without the independence property), giving several characterizations, answering some questions from
ASSOCIATIVITY OF THE MORLEY PRODUCT OF INVARIANT MEASURES IN NIP THEORIES
TLDR
A new proof of associativity for the Morley (or “nonforking”) product of invariant measures in NIP theories is given.
Definable convolution and idempotent Keisler measures
We initiate a systematic study of the convolution operation on Keisler measures, generalizing the work of Newelski in the case of types. Adapting results of Glicksberg, we show that the supports of
Sequential approximations for types and Keisler measures
TLDR
It is shown that both generically stable types and Keisler measures which are finitely satisfiable over a countable model (in NIP theories) are sequentially approximated.
Definable groups and compact p‐adic Lie groups
We formulate p‐adic analogues of the o‐minimal group conjectures from the works of Hrushovski, Peterzil and Pillay [J. Amer. Math. Soc., to appear] and Pillay [J. Math. Log. 4 (2004) 147–162]; that
...
...