We develop “local NIP group theory” in the context of pseudofinite groups. In particular, given a sufficiently saturated pseudofinite structure G expanding a group, and left invariant NIP formula δ(x; ȳ), we prove various aspects of “local fsg” for the right-stratified formula δr(x; ȳ,u) := δ(x ⋅ u; ȳ). This includes a δr-type-definable connected component, uniqueness of the pseudofinite counting measure as a left-invariant measure on δr-formulas and generic compact domination for δr-definable sets.

中文翻译：

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赝有限群和 VC 维

我们在伪有限群的背景下发展了“局部 NIP 群论”。特别是，给定一个足够饱和的伪有限结构G展开一个群，并留下不变的 NIP 公式δ(X; )，我们证明了“局部fsg” 为右分层公式δr(X; ,你) ：= δ(X ⋅ 你; ). 这包括一个δr-类型可定义的连接组件，伪有限计数度量的唯一性作为左不变度量δr- 公式和通用紧凑控制δr- 可定义的集合。