Removing mild assumptions in the NIP ω-categorical classification of invariant Keisler measures

Determine whether the classification of invariant Keisler measures for ω-categorical NIP theories—namely, that every invariant Keisler measure is an integral average of Aut(M/acl^{eq}(∅))-invariant types—holds without the additional assumption that no Aut(M/acl^{eq}(∅))-invariant type has uncountably many Aut(M)-conjugates. Equivalently, remove the auxiliary mild assumptions used in Section 7 to obtain the decomposition result.

Background

In Section 7 the authors provide a decomposition of invariant Keisler measures in ω-categorical settings under mild assumptions (in particular, using IEP—which they show holds for NIP ω-categorical theories—and a restriction on the size of Aut(M)-conjugacy classes of Aut(M/acl^{eq}(∅))-invariant types).

They explicitly note that it remains open whether these auxiliary assumptions are necessary, i.e., whether the decomposition of invariant Keisler measures into weighted averages of invariant types can be established for all NIP ω-categorical theories without further conditions.

Resolving this would yield a complete characterization of invariant Keisler measures in the NIP ω-categorical context.