## Notre Dame Journal of Formal Logic

### 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 $\mu(\phi(x,b))=0$ for all $b$, then for some $n$, $\mu^{(n)}(\exists y(\phi(x_{1},y)\wedge \cdots \wedge\phi(x_{n},y)))=0$. 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 $\emptyset$, precisely as in stable groups, answering positively an earlier problem posed by the first two authors.

#### Article information

Source
Notre Dame J. Formal Logic Volume 53, Number 4 (2012), 599-605.

Dates
First available in Project Euclid: 8 November 2012

https://projecteuclid.org/euclid.ndjfl/1352383235

Digital Object Identifier
doi:10.1215/00294527-1814705

Mathematical Reviews number (MathSciNet)
MR2995423

Zentralblatt MATH identifier
1318.03046

Keywords
Keisler measures NIP dependent fsg

#### Citation

Hrushovski, Ehud; Pillay, Anand; Simon, Pierre. A Note on Generically Stable Measures and fsg Groups. Notre Dame J. Formal Logic 53 (2012), no. 4, 599--605. doi:10.1215/00294527-1814705. https://projecteuclid.org/euclid.ndjfl/1352383235.

#### References

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• [3] Hrushovski, E., and A. Pillay, “On $\mathit{NIP}$ and invariant measures,” Journal of the European Mathematical Society, vol. 13 (2011), pp. 1005–61.
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