Measurable Local Lemma for Arbitrary Group Actions

Definitions and mathematical statement

Let $\Gamma$ be a group and let $(X,\mathcal{B},\mu)$ be a probability space. A probability-measure-preserving action $\Gamma\curvearrowright X$ consists of measurable bijections $T_g:X\to X$, indexed by $g\in\Gamma$, satisfying \[ T_e=\operatorname{id}_X,\qquad T_{gh}=T_g\circ T_h,\qquad \mu(T_g^{-1}(A))=\mu(A) \quad(g,h\in\Gamma,\ A\in\mathcal{B}). \] The supplied question asks whether the measurable local lemma holds for every such action of every group, without an explicit freeness restriction.

Unresolved formulation. The record does not define the version of the measurable local lemma intended here. In particular, it does not specify the admissible constraint systems or bad events, their dependency relation, the probability bounds required, the measurable object to be constructed, or whether constraints must be satisfied everywhere or almost everywhere. It also does not specify whether groups must be countable, whether probability spaces must be standard, or whether further hypotheses on actions are implicit. Consequently, the universal question can be preserved, but a precise quantified local-lemma assertion cannot be recovered from this record alone. These missing specifications require source clarification and human mathematical review; no particular local-lemma criterion is adopted in this draft.

Short English statement

Does the measurable local lemma hold for every probability-measure-preserving action of every group? The record leaves the precise local-lemma formulation and its ambient hypotheses unspecified, so the question remains a statement-only draft requiring clarification.

Sources

• Supplied local UnsolvedMath registry record, permanent ID $30003215$, title Measurable Local Lemma for Arbitrary Group Actions, source problem code catalogue entry, in the Oberwolfach Reports Open Problems collection. The record gives proposed year $2016$ and names no individual proposer.
• Supplied report citation: Measured Group Theory (2017), Oberwolfach Reports, pp. 2347--2397, DOI 10.4171/owr/2016/41. The background describes these reports as documenting mathematical workshops and their open-problem discussions. Original open problem: https://doi.org/10.4171/owr/2016/41. The relationship between the citation's year $2017$ and the record's year $2016$ is not explained.
• Supplied literature reference labeled "Bernshteyn": https://doi.org/10.1016/j.aim.2019.06.031. No title, theorem number, or precise formulation is supplied for this reference.
• Literature assessment dated exactly 2026-08-21: the supplied curation labels the problem open and reports that the measurable local lemma is established for Bernoulli actions, essentially free amenable actions, and other restricted classes, while the formulation for all groups and all probability-measure-preserving actions remains unresolved. This is supplied dated curation derived from the OWR material, not an independently verified assessment. The background explicitly requests independent verification of time-sensitive status and cited claims.
• The registry labels the statement corrected_verified and reports manual semantic, typographic, and LaTeX review asserting that it is self-contained and preserves the mathematical target. This supplied review label does not resolve the formulation omissions identified above; the cited sources have not been visited or independently checked for this draft.