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A concept in the theory of harmonic functions (cf. [[Harmonic function|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 [[#References|[1]]], [[#References|[2]]]. 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|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|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|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|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|Poisson integral]]). This is the reason for the importance of the various estimation methods for harmonic measure [[#References|[4]]], [[#References|[5]]], [[#References|[6]]], [[#References|[7]]], which are mainly based on the principle of extension of domain (cf. [[Extension of domain, principle of|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.

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  T. Carleman,   "Sur les fonctions inverses des fonctions entières d'ordre fini"  ''Ark. Mat.'' , '''15''' :  10  (1921)  pp. 1–7</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  F. Nevanlinna,   R. Nevanlinna,   "Ueber die Eigenschaften einer analytischen Funktion in der Umgebung einer singulären Stelle oder Linie"  ''Acta Soc. Sci. Fennica'' , '''50''' :  5  (1922)  pp. 1–46</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  Ch.J. de la Vallée-Poussin,   ''Ann. Inst. H. Poincaré'' , '''2'''  (1932)  pp. 169–232</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  R. Nevanilinna,   "Analytic functions" , Springer  (1970)  (Translated from German)</TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top">  G.M. Goluzin,   "Geometric theory of functions of a complex variable" , ''Transl. Math. Monogr.'' , '''26''' , Amer. Math. Soc.  (1969)  (Translated from Russian)</TD></TR><TR><TD valign="top">[6]</TD> <TD valign="top">  M. Brélot,   "Eléments de la théorie classique du potentiel" , Sorbonne Univ. Centre Doc. Univ. , Paris  (1959)</TD></TR><TR><TD valign="top">[7]</TD> <TD valign="top">  K. Haliste,   "Estimates of harmonic measure"  ''Ark. Mat.'' , '''6''' :  1  (1965)  pp. 1–31</TD></TR></table>

====Comments====
Harmonic measure is an important tool in axiomatic potential theory (cf. [[Potential theory, abstract|Potential theory, abstract]]), see [[#References|[a1]]].

For domains in  $  \mathbf C $
very precise estimates for harmonic measure in terms of [[Hausdorff measure|Hausdorff measure]] have been obtained recently. Let  $  h( t) $
be a continuous increasing function for  $  t \geq  0 $, 
$  h( 0)= 0 $, 
and let  $  E $
be a [[Borel set|Borel set]]. Let  $  \Lambda _ {h} ( E) $
denote the Hausdorff measure with respect to  $  h $
of  $  E $. 
Let  $  D $
be a domain in  $  \mathbf C $
and set  $  \omega ( E) = \omega ( x;  E, D) $. 
Makarov's theorems [[#References|[a3]]] are: 1) Suppose that  $  D $
is simply connected. If  $  \lim\limits _ {t \rightarrow \infty }   h( t) /t = 0 $, 
then  $  \omega $
is singular with respect to  $  \Lambda _ {h} $, 
i.e.  $  \omega \perp  \Lambda _ {h} $. 
2) There exist constants  $  C _ {1} $, 
$  C _ {2} $
with the following property: Let

$$ 
h _ {i} ( t)  =  t   \mathop{\rm exp} \left \{ C _ {i} \sqrt {\left (  \mathop{\rm log}  
\frac{1}{t}
 \right )   \mathop{\rm log}   \mathop{\rm log}   \mathop{\rm log}  
\frac{1}{t}
 }
\right \} ,\  i = 1, 2.
$$

Then for every Jordan domain  $  D $, 
$  \omega $
is absolutely continuous with respect to  $  \Lambda _ {h _ {1}  } $, 
i.e.  $  \omega \ll  \Lambda _ {h _ {1}  } $. 
However, there exists a Jordan domain  $  D $
with  $  \omega \perp  \Lambda _ {h _ {2}  } $.

Next (B. Øksendal, Jones, Wolff): If  $  1 < a \leq  2 $, 
then for every domain  $  D $
in  $  \mathbf C $, 
$  \omega \perp  \Lambda _ {( t  ^  \alpha   ) } $.

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  C. Constantinescu,   A. Cornea,   "Potential theory on harmonic spaces" , Springer  (1972)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  J.B. Garnett,   "Applications of harmonic measure" , Wiley (Interscience)  (1986)</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  N. Makarov,   "On the distortion of boundary sets under conformal mappings"  ''Proc. London Math. Soc.'' , '''51'''  (1985)  pp. 369–384</TD></TR></table>