Examples
- If is the unit disk, then harmonic measure of with pole at the origin is length measure on the unit circle normalized to be a probability, i.e. for all where denotes the length of .
- If is the unit disk and, then for all where denotes length measure on the unit circle. The Radon-Nikodym derivative is called the Poisson kernel.
- More generally, if and is the n-dimensional unit ball, then harmonic measure with pole at is for all where denotes surface measure (-dimensional Hausdorff measure) on the unit sphere and .
- If is a simply connected planar domain bounded by a Jordan curve and XD, then for all where is the unique Riemann map which sends the origin to X, i.e. . See Carathéodory's theorem.
- If is the domain bounded by the Koch snowflake, then there exists a subset of the Koch snowflake such that has zero length and full harmonic measure .
Read more about this topic: Harmonic Measure
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