Subharmonic Functions

Abstract

In this section we introduce a class of functions that generalize to the case of an arbitrary system [Σ, a] the familiar plurisubharmonic functions associated with [ℂⁿ, ℘]. The functions considered belong to the larger class U of all upper semicontinuous (use) functions defined on arbitrary subsets of Σ with values in the extended real numbers [-∞, ∞). Thus, an element f ∈ U may assume the value -∞ but not the value +∞ and, for each point 6 in its domain of definition, $\mathop{{\lim \,\sup }}\limits_{{\sigma \to \delta }} \,f(\sigma ) \leqslant f(\delta )$. Note that U is closed under multiplication by positive reals and under restrictions, i.e., if f ∈ U and X is a subset of the domain of f then also f|X ∈ U. Furthermore, if f, g ∈ U then f + g ∈ U provided f + g exists (i.e. the domains of f and g intersect). With the obvious convention, we accordingly say that U is “closed under linear combinations with positive coefficients.”