A subset E of the boundary ∂Ω of a domain Ω in which the harmonic measure ω^x(E) equals zero for a chosen reference interior point x (or for harmonic measure from every interior point); equivalently, E is charged with zero probability by Brownian motion started at x upon first exit of Ω.

Principle

Harmonic measure encodes how harmonic functions on Ω take boundary values and how Brownian paths exit Ω; a harmonic-measure-zero subset carries no boundary mass for the Poisson integral and is negligible for harmonic boundary behavior relative to the chosen basepoint.

Demonstration

On the unit disk, a closed subset of the circle of Lebesgue measure zero can be arranged (by conformal mapping or construction) to have harmonic measure zero relative to the center; Brownian motion started at the center almost surely exits the disk at points outside that subset.

Misapplication

Equating harmonic-measure-zero with other smallness notions such as Lebesgue measure zero or Hausdorff dimension zero in all contexts; harmonic measure reflects conformal/potential-theoretic geometry and can differ substantially from these other measures.

Consequence

If a boundary set has harmonic measure zero, bounded harmonic functions are unaffected by prescribing arbitrary bounded values on that set (they do not change the Poisson extension), and probabilistically the set is almost never the exit location of Brownian motion from the domain.

Reversal

A set of positive harmonic measure: a boundary subset that receives positive mass under harmonic measure and thus influences harmonic extensions and Brownian exit distributions.

Boundary

Depends on the domain Ω and the choice of basepoint for harmonic measure; harmonic-measure-zero is not an intrinsic property of the abstract set detached from the embedding as part of ∂Ω and the potential-theoretic context.

Semantic Tension

The tension is between measure-theoretic smallness (Lebesgue or Hausdorff) and potential-theoretic smallness (harmonic measure or capacity); sets small in one sense need not be small in the other, so context matters.

Synthesis

A harmonic-measure-zero set is a boundary subset that carries no mass for the domain's harmonic measure, meaning it is negligible for Poisson representation of harmonic functions and almost surely not hit by Brownian exit, yet its smallness must be judged relative to the domain and potential-theoretic geometry.