  We investigate various forms of (model-theoretic) stability for hypergraphs
and their corresponding strengthenings of the hypergraph regularity lemma with
respect to partitions of vertices. On the one hand, we provide a complete
classification of the various possibilities in the ternary case. On the other
hand, we provide an example of a family of slice-wise stable 3-hypergraphs so
that for no partition of the vertices, any triple of parts has density close to
0 or 1. In particular, this addresses some questions and conjectures of Terry
and Wolf. We work in the general measure theoretic context of graded
probability spaces, so all our results apply both to measures in ultraproducts
of finite graphs, leading to the aforementioned combinatorial applications, and
to commuting definable Keisler measures, leading to applications in model
theory.
