Invariant measures in simple and in small theories. (English) Zbl 07681055
Summary: We give examples of (i) a simple theory with a formula (with parameters) which does not fork over \(\emptyset\) but has \(\mu\)-measure 0 for every automorphism invariant Keisler measure \(\mu\) and (ii) a definable group \(G\) in a simple theory such that \(G\) is not definably amenable, i.e. there is no translation invariant Keisler measure on \(G\). 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.
MSC:
| 03-XX | Mathematical logic and foundations |
Keywords:
Keisler measure; simple theory; definable group; definable amenability; paradoxical decomposition; Grothendieck ringReferences:
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