Invariant measures in simple and in small theories
Artem Chernikov, Ehud Hrushovski, Alex Kruckman, Krzysztof Krupiński, Slavko Moconja, Anand Pillay, Nicholas Ramsey
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Source: Crossref
Published: Dec 17, 2022
DOI: 10.1142/s0219061322500258
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We give examples of (i) a simple theory with a formula (with parameters) which does not fork over [Formula: see text] but has [Formula: see text]-measure 0 for every automorphism invariant Keisler measure [Formula: see text] and (ii) a definable group [Formula: see text] in a simple theory such that [Formula: see text] is not definably amenable, i.e. there is no translation invariant Keisler measure on [Formula: see text]. We also discuss paradoxical decompositions both in the setting of discrete groups and of definable groups, and prove some positive results about small theories, including the definable amenability of definable groups.
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