By J. L. Doob (auth.)

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Chapter III Infima of Families of Superharmonic Functions 1. Least Superharmonic Majorant (LM) and Greatest Subharmonic Minorant (GM) If r is a class of extended real-valued functions on a set D, a function v on D will be called a majorant [minorant] of r if v ~ u [v ~ u] for every u in r. If D is an open subset of IR N , the least superharmonic majorant [greatest subharmonic minorant] of r, if such a function exists, will be denoted by LMDr [GMDr], or by LMDu [GMDu] ifr = {u} is a singleton. If a superharmonic function u on D has a subharmonic minorant, then GMDu exists and is harmonic.

Basic Properties of Harmonic, Subharmonic, and Superharmonic Functions 10gM(/z/). Since n-+L(vn' 0, r) is a convex function of log r, the same is true of the function r ~ log M(r). See a slightly different derivation of this same property of M(') in the context of the minimum function of a superharmonic function in Section 10. Alternatively, the continuous function v is subharmonic because it has the sub harmonic function average property and therefore the function r~logM(r) = L(v, 0, r) is a convex function oflog r.

L the Green function GD will be defined for every Greenian set D. The property GMDGD(~") = 0 remains true and in fact is very nearly a defining property ofG D· 2. Generalization of Theorem 1 Theorem 1 is included in the following theorem but was proved separately because of the importance of its constructive proof. Theorem. If a class r of superharmonic functions on an open subset D of [RN has a subharmonic minorant, then GMr exists and is harmonic. Let robe the class of subharmonic minorants of r.