By L. I. Ronkin
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Additional resources for Introduction to the theory of entire functions of several variables
Example text
Properties (a) and (b) are clearly valid for arbitrary sets, for both inner and outer capacity. e. if Cap A = Cap A, the set A is said to be capacitable, and the common value of the inner and outer capacities of A is called its capacity, denoted as before by Cap A (Capm A): Cap A= Cap A= Cap A. It is readily seen that closed and open sets are capacitable. 1 (CHOQUET (1)). 17> The capacities of Borel sets possess properties (a)-(e) enumerated above for compact sets. To end this subsection, we mention another important property of the capacity of a compact set, of a somewhat different nature from properties (a)- (e).
Integral representation. We first show that every measure µ on a domain G c: Rm such that µ( G) < oo is associated with at least one subharmonic function u(x) such that L1u/(}m = µ. 1). e. (L1h, rp) = Omrp(O) for all rp e D(Rm). Indeed, and h(x) L1h = (L1h, 2. ¢ + u2)ml2 = 1 + ~1 B':+2(¢)vn, where each term B':+ 2 (ifi)vn is a homogeneous polynomial of degree n in the coordinates of x. Substituting this expression into the Poisson integral, we get a power series expansion of the function