# Invariant measures in simple and in small theories

Chernikov, A
Hrushovski, E
Kruckman, A
Krupinski, K
Moconja, S
Pillay, A
Ramsey, N

## Last Updated:

2021-08-23T19:45:56.303+01:00

## abstract:

We give examples of (i) a simple theory with a formula (with parameters)
which does not fork over the empty set 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, and
nontriviality of the graded Grothendieck ring.

1179506

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Journal Article