Chaining and Weak Mixing for Measure-Class-Preserving Actions

Definitions and mathematical statement

For a group $G$ acting by measurable bijections with measurable inverses on a measure space $(X,\Sigma,\mu)$, the action preserves the measure class if \[ \mu(A)=0\quad\Longleftrightarrow\quad\mu(gA)=0 \qquad(g\in G,\ A\in\Sigma). \] The supplied record does not specify the intended class of groups, measure spaces, or actions. It also does not define chaining, $1$-chaining, or the convention for weak mixing for measure-class-preserving actions. These missing specifications prevent a self-contained precise formulation.

Subject to those unresolved specifications, let $\mathcal A$ denote the intended class of measure-class-preserving actions, and let $\operatorname{Ch}(\alpha)$, $\operatorname{Ch}_1(\alpha)$, and $\operatorname{WM}(\alpha)$ denote the three properties in the intended source definitions. These symbols are placeholders, not definitions of those properties. The two supplied questions are:

1. Is the following universal equivalence true? \[ \forall\alpha\in\mathcal A,\qquad \operatorname{Ch}(\alpha)\ \Longleftrightarrow\ \operatorname{WM}(\alpha). \]
2. Does the following existential statement hold? \[ \exists\alpha\in\mathcal A,\qquad \operatorname{Ch}(\alpha)\ \land\ \neg\operatorname{Ch}_1(\alpha). \]

The supplied background attributes to the report the equivalence of chaining, weak mixing, and $1$-chaining for probability-preserving actions, and says that both measure-class-preserving questions remain unresolved. This is supplied contextual information; the definitions and hypotheses of that probability-preserving result are not included.

Short English statement

The source asks whether chaining and weak mixing coincide for every action in the intended measure-class-preserving setting, and whether some such action is chaining without being $1$-chaining. Definitions of the three properties and the precise class of actions must be recovered before this draft can be made self-contained. The supplied record labels both questions open.

Sources

• Supplied local UnsolvedMath registry record: permanent ID $30006170$, source problem code catalogue entry, title Chaining and Weak Mixing for Measure-Class-Preserving Actions; collection: Oberwolfach Reports Open Problems. No individual proposer is identified; the supplied proposed year is 2025.
• Supplied report citation: Set Theory (2025), Oberwolfach Reports, pp. 75--140, DOI 10.4171/owr/2025/2. https://doi.org/10.4171/owr/2025/2.
• Supplied literature source: Oberwolfach Report 2/2025. https://content.ems.press/assets/public/full-texts/serials/owr/22/1/14299082/online/10.4171-owr-2025-2.pdf.
• Supplied literature source: Tserunyan--Zomback, A backward ergodic theorem along trees. https://arxiv.org/abs/2012.10522.
• Literature assessment dated exactly 2026-08-22: the supplied curation labels the questions open and states that the report proves equivalence of chaining, weak mixing, and one-chaining for probability-preserving actions while leaving both measure-class-preserving questions unresolved. This is supplied dated curation derived from the OWR record, not an independently verified literature assessment; its status and cited claims require independent verification.
• The registry labels the statement corrected_verified and says that it was reconstructed and checked against the official report. Those are supplied verification claims, not verification performed for this draft.