关于幂等测度猜想和不变测度的分解
On idempotent measure conjecture and decomposition of invariant measures
AI总结：
本文通过*积和f-通用类型研究不变类型的最小理想，并分析幂等测度猜想在可去NIP理论中的分类与应用。
摘要：
我们利用[GHK25]中引入的*积和f-通用类型来描述不变类型的最小理想，并对可去NIP理论中的测度进行分类。此外，我们分析了在[CGK24]和[GHK25]中研究的所谓幂等测度猜想周围的情况。
英文摘要：
We continue the study of the semigroup of global invariant types introduced by Gannon, Hoffmann, and Krupiński and the associated convolution semigroup of invariant Keisler measures. The first part of the paper concerns the Idempotent Measure Conjecture, studied in [CGK24] and [GHK25], which predicts that idempotent fim Keisler measures should be precisely the invariant Haar measures on relatively type-definable subgroups. We identify structural properties underlying all previously known positive instances of the conjecture and prove a conditional version under natural semigroup-theoretic hypotheses. In particular, we show that left simplicity of the support and a local invariance condition are sufficient for the conjectural characterization. The second part studies invariant Keisler measures in amenable NIP theories. We prove that the semigroup of global invariant types has a unique minimal left ideal consisting of f-generic types and identify its Ellis groups. Using normalized Haar measure on the Kim-Pillay group of T, we associate canonically an invariant Keisler measure to every Ellis group. We show that the resulting measures are supported on minimal subflows, prove that the flow (S_m(C), Aut(C)) is hereditarily amenable, establish unique ergodicity of every minimal subflow in the countable case, and characterize all ergodic Aut(C)-invariant Keisler measures as the measures arising from this construction.
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