A Note on Generically Stable Measures and fsg Groups

We prove (Proposition 2.1) that if μ is a generically stable measure in an NIP theory, and μ(φ(x, b)) = 0 for all b then for some n, μ(∃y(φ(x1, y)∧ ..∧φ(xn, y))) = 0. As a consequence we show (Proposition 3.2) that if G is a definable group with fsg 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 ∅, precisely as in stable groups, answering positively Problem 5.5 from [3].