Existence Criteria for $g$-Measures

Definitions and mathematical statement

Supplied question. What conditions on the size or shape of the state set $D$ ensure the existence of a $g$-measure?

Unresolved definitions and quantifiers. The supplied record calls $D$ a state set but specifies neither its ambient space nor its measurable, topological, or geometric structure. It does not define $g$, its domain, its normalization or regularity assumptions, or the meaning of a $g$-measure. Consequently, neither "size" nor "shape" has a specified mathematical meaning here. It is also unspecified whether $g$ is fixed, ranges over a prescribed class, or is to be chosen together with the measure. These omissions prevent a precise quantified existence statement. The faithful target is therefore the classification question above, with the definitions, hypotheses, and quantifiers still requiring recovery from the underlying source; no particular interpretation of $D$ or convention for $g$-measures is imposed.

Short English statement

Determine which size or shape conditions on $D$ guarantee a $g$-measure. The supplied extract leaves the state space, the function $g$, and the intended compatibility conditions undefined, so a precise formulation requires further source clarification. The supplied curation labels this an open classification problem.

Sources

• Supplied local UnsolvedMath registry record, permanent ID $30004430$, source problem code catalogue entry, titled "Existence Criteria for $g$-Measures," in the Oberwolfach Reports Open Problems collection. No individual proposer is identified.
• Supplied report citation: Mini-Workshop: One-sided and Two-sided Stochastic Descriptions (2021), Oberwolfach Reports, pp. 601--637, DOI 10.4171/owr/2020/11. https://doi.org/10.4171/owr/2020/11. The record gives proposed year 2020; the citation's year 2021 and the 2020 identifiers are retained without reconciliation.
• Supplied literature source, labeled OWR-2020-11: https://ems.press/content/serial-article-files/46847.
• Literature assessment dated 2026-08-22: the supplied curation describes the question as an open classification problem asking for general size and shape conditions on $D$, rather than a single resolved theorem. This is supplied dated curation, not an independently verified literature assessment. The record also labels the statement corrected_verified and reports a manual semantic, typographic, and LaTeX review; those are supplied review claims and do not resolve the missing mathematical definitions.