Research Problems in Function Theory — Problem 7.50

Definitions and mathematical statement

For a connected Euclidean-open set $U\subset{\mathbb{R}}^n$, define the rectifiably accessible boundary set by \[ A_{\mathrm{rect}}(U)=\{\xi\in\partial U:\exists\gamma:[0,1]\to{\mathbb{R}}^n\text{ continuous},\ \gamma([0,1))\subset U,\ \gamma(1)=\xi,\ \operatorname{length}(\gamma)<\infty\}. \] The source attributes to Brelot--Choquet the assertion that harmonic measure is carried by this set. In measure-theoretic terms, for each applicable interior pole $x$, this means $\omega_U^x(\partial U\setminus A_{\mathrm{rect}}(U))=0$, rather than a claim about the topological support being a subset of the accessible set.

The fine topology is the potential-theoretic topology generated to make superharmonic functions continuous; a finely open set is open in this topology. T. Lyons and B. Øksendal ask whether the analogous assertion holds for finely open sets and Keldysh measure: is the relevant boundary measure carried by boundary points accessible from inside by finite-length rectifiable paths?

The source does not define Keldysh measure, its pole and existence assumptions, which boundary is used for a finely open set, or whether the path/end-point topology remains Euclidean or is fine. It also does not say whether ordinary connectedness is to be replaced by fine connectedness, or which dimension/potential-theory conventions are intended. The cited motivation about polygonal path connectedness cannot silently remove a necessary connectedness qualification. These choices are required before writing a unique fine-potential-theoretic measure identity; no standard harmonic measure on an ordinary open domain is substituted for Keldysh measure.

Short English statement

Does the finite-rectifiable-path accessibility theorem for harmonic measure extend to fine potential theory and Keldysh measure? The source invokes polygonal path connectedness as motivation, but omits the precise fine-domain, boundary, path and measure conventions. The supplied triage reports openness in Hayman's 2018 update and no definitive resolution located; no independent current-status review is asserted.

Sources

• UnsolvedMath permanent ID 2307050, source code catalogue entry, "Research Problems in Function Theory --- Problem 7.50." Registry curation and metadata: Ulam AI and UnsolvedMath Contributors, CC BY 4.0; updated 2026-08-06.
• Hayman, Research Problems in Function Theory (2018 source update), Problem 7.50, attributed to T. Lyons and B. Øksendal: https://arxiv.org/abs/1809.07200. Supplied source-TeX extraction accessed 2026-07-29. The metadata retains the update for status triage and marks status/release and rights review NEEDS_REVIEW.
• Brelot and Choquet are named for the ordinary-domain harmonic-measure theorem; no complete reference is supplied for it, the fine path-connectedness motivation, or Keldysh measure. No missing citation is inferred.