Research Problems in Function Theory — Problem 6.115

Definitions and mathematical statement

Shared notation.

$\mathbb N=\{1,2,\ldots\}$, and $\mathbb Z,\mathbb Q,\mathbb R,\mathbb C$ denote the integers, rationals, reals and complex numbers. Any other conventions are specified locally.

For a proper plane domain $U$, a Borel set $E\subseteq\partial U$, and $z\in U$, $\omega(z,E,U)$ denotes harmonic measure: the probability that planar Brownian motion begun at $z$ first exits $U$ through $E$. For an arbitrary set, use the associated outer measure; for sets off the boundary, use their intersection with $\partial U$. A rectifiable curve $\Gamma\subseteq\mathbb C$ has finite length $\mathcal H^1(\Gamma)<\infty$. Let $E\subseteq\Gamma$ satisfy $\mathcal H^1(E)=0$, and let $\Omega\subsetneq\mathbb C$ be any simply connected domain with $z\in\Omega$.

Statement recorded in the supplied source.

Must $\omega(z,E\cap\partial\Omega,\Omega)=0$ hold? The curve need not coincide with the domain boundary. For $\Omega=\mathbb C$, the boundary is empty and the asserted measure is zero. [S2]

Short English statement

Is every length-zero subset of a rectifiable curve negligible for harmonic measure of every simply connected domain?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 2306115 (AMR-022-6115). CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 W. K. Hayman and E. F. Lingham, Research Problems in Function Theory (2018), Problem 6.115, complete problem and update. https://arxiv.org/abs/1809.07200.