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Support transitivity of idempotent generically stable measures

Show that every generically stable idempotent global Keisler measure μ on a type‑definable group G(U) is support transitive; that is, prove μ ∗ p = μ for all types p in the support S(μ).

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Background

The authors introduce support transitivity as a weaker property than full generic transitivity for measures and note it holds trivially for idempotent types. Establishing it for measures would advance the program of transferring type‑based arguments via randomization.

They suggest that adapting type‑level techniques may reduce the main conjecture on generic transitivity to this weaker property, but the status of support transitivity itself remains unresolved.

References

It leads to a weaker conjecture saying that every generically stable idempotent measure is support transitive (see Problem \ref{intermediate:conjecture}). While this conjecture is open, it trivially holds for idempotent types, and so one can expect that if the techniques used for types in Sections \ref{subsection: stable theories}--\ref{sec: rosy types} could be adapted to measures, they would rather not prove the main conjecture that every generically stable idempotent measure is generically transitive, but reduce it to the above weakening.

— Definable convolution and idempotent Keisler measures III. Generic stability, generic transitivity, and revised Newelski's conjecture (2406.00912 - Chernikov et al., 3 Jun 2024) in Section 3.7 (Support transitivity of idempotent measures)