We give examples of (i) a simple theory with a formula (with parameters) which does not fork over $\varnothing$ 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.

中文翻译：

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简单和小理论中的不变测度

我们给出了 (i) 一个带有公式（带有参数）的简单理论的例子，它不会分叉$\varnothing$但是有$\mu$- 对每个自同构不变的 Keisler 测度测量 0$\mu$(ii) 一个可定义的组$G$在一个简单的理论中$G$不是绝对服从的，即不存在平移不变的凯斯勒测度$G$. 我们还讨论了离散群和可定义群的设置中的悖论分解，并证明了一些关于小理论的积极结果，包括可定义群的可定义适用性。