# Harmonic measure

A concept in the theory of harmonic functions (cf. Harmonic function) connected with estimating the modulus of an analytic function inside a domain if certain bounds on the modulus on the boundary of the domain are known , . Let $D$ be a bounded open set in the Euclidean space $\mathbf R ^ {n}$, $n \geq 2$; let $\Gamma = \partial D$ be the boundary of $D$; and let $f$ be a finite real-valued continuous function on $\Gamma$. To each such function $f$ there corresponds a unique harmonic function $H _ {f} ( x)$ on $D$ which, for $f$, is a generalized solution of the Dirichlet problem. If the point $x \in D$ is assumed to be fixed, the functional $H _ {f} ( x)$ will define on the compact set $\Gamma$ a positive Radon measure $\omega ( x) = \omega ( x, D)$, which is called the harmonic measure at the point $x$. The formula for the representation of the generalized solution of the Dirichlet problem,

$$H _ {f} ( x) = \ \int\limits f ( y) d \omega ( x; D),$$

obtained by Ch.J. de la Vallée-Poussin by the balayage method, is valid for all functions $f$ which are continuous on $\Gamma$. Moreover, if $E$ is an arbitrary Borel set on $\Gamma$, the harmonic measure $\omega ( x; E, D)$, $x \in D$, of $E$ at $x$ is equal to the value at $x$ of the generalized solution of the Dirichlet problem for the characteristic function $\chi _ {E} ( y)$, $y \in \Gamma$, of $E$.

The basic properties of a harmonic measure are: $\omega ( x; E, D)$ is a harmonic function of the point $x$ in $D$;

$$0 \leq \omega ( x; E, D) \leq 1;$$

$$1 - \omega ( x; E, D) = \omega ( x; \Gamma \setminus E, D);$$

if $D$ is a domain and $\omega ( x; E, D) = 0$ even at a single point $x \in D$, then $\omega ( x; E, D) \equiv 0$.

In the last-named case $E$ is known as a set of harmonic measure zero. If a compact set $K \subset \mathbf R ^ {n}$ has harmonic measure zero with respect to some domain $D$, $K \subset D$, i.e. $\omega ( x; K, D \setminus K) = 0$, then it has harmonic measure zero with respect to all other domains, i.e. $K$ is a set of absolute harmonic measure zero. A set $K$ has absolute harmonic measure zero if and only if it has zero (harmonic) capacity.

As regards applications to the theory of functions of a complex variable, the dependence of a harmonic measure on the domain $D$ is of special importance. This dependence is expressed by the principle of harmonic measure (cf. Harmonic measure, principle of), which states that a harmonic measure does not decrease under mappings of the domain $D$ realized by univalent analytic functions $w = w( z)$, $z \in D$. In particular, a harmonic measure remains unchanged under a one-to-one conformal mapping.

Explicit computations of harmonic measures are possible only for the simplest domains $D$( mainly for the disc, for the sphere, for a half-plane, and for a half-space; see Poisson integral). This is the reason for the importance of the various estimation methods for harmonic measure , , , , which are mainly based on the principle of extension of domain (cf. Extension of domain, principle of). In the simplest form, for $n = 2$, this principle consists in the following: Let a finitely-connected domain $D$ be bounded by a finite number of Jordan curves $\Gamma$ and let $\alpha$ be an arc on $\Gamma$. Then, if the domain $D$ is extended in some way across the complementary part $\Gamma \setminus \alpha$ of the boundary, the harmonic measure $\omega ( z; \alpha , D)$ can only increase.

How to Cite This Entry:
Harmonic measure. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Harmonic_measure&oldid=47182
This article was adapted from an original article by E.D. Solomentsev (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article