INVARIANT MEASURES IN SIMPLE AND IN SMALL THEORIES

. We give examples of (i) a simple theory with a formula (with parameters) which does not fork over ∅ but has µ -measure 0 for every automorphism invariant Keisler measure µ , 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.

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