A Note on Generically Stable Measures and fsg Groups

Abstract

We prove (Proposition 2.1) that if $\mu$ is a generically stable measure in an NIP (no independence property) theory, and $\mu \left(\varphi \right(x,b\left)\right)=0$ for all $b$, then for some $n$, ${\mu }^{\left(n\right)}\left(\exists y\right(\varphi (x_1,y)\wedge \cdots \wedge \varphi (x_n,y)\left)\right)=0$. As a consequence we show (Proposition 3.2) that if $G$ is a definable group with fsg (finitely satisfiable generics) in an NIP theory, and $X$ is a definable subset of $G$, then $X$ is generic if and only if every translate of $X$ does not fork over $\varnothing$, precisely as in stable groups, answering positively an earlier problem posed by the first two authors.