Subharmonic functions and associated measures in ℝ ^<i>n</i>

Abstract For subharmonic functions <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi>s</m:mi> </m:math> s in <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msup> <m:mrow> <m:mi mathvariant="double-struck">R</m:mi> </m:mrow> <m:mrow> <m:mi>n</m:mi> </m:mrow> </m:msup> <m:mo>,</m:mo> <m:mi>n</m:mi> <m:mo>≥</m:mo> <m:mn>2</m:mn> <m:mo>,</m:mo> </m:math> {{\mathbb{R}}}^{n},n\ge 2, there is an associated Radon measure <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi>μ</m:mi> </m:math> \mu that is used to represent <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi>s</m:mi> </m:math> s locally as an integral up to an additive harmonic function. We prove that the total measure <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi>μ</m:mi> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mrow> <m:msup> <m:mrow> <m:mi mathvariant="double-struck">R</m:mi> </m:mrow> <m:mrow> <m:mn>2</m:mn> </m:mrow> </m:msup> </m:mrow> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:math> \mu ({{\mathbb{R}}}^{2}) is finite if and only if the subharmonic function <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi>s</m:mi> </m:math> s has harmonic majorant outside a compact set. However, this relationship does not hold in higher dimensions.