= Perron family of subharmonic functions
{c}
A nonempty collection of real continuous <subharmonic functions> on a connected <Riemann surface> is a Perron family if it is closed under pairwise maxima and <harmonic lifting of a continuous subharmonic function> on every relatively compact coordinate disk. Its pointwise supremum is either everywhere infinite or a <harmonic function on a Riemann surface>. At a point where the supremum is finite, take increasing disk lifts approaching that value. The <Harnack inequality for harmonic functions> gives a finite harmonic limit on the disk. Lifting their maxima with any other member gives a larger harmonic limit with the same value at the chosen point, hence the same limit by the <strong maximum principle for harmonic functions>. The supremum is therefore harmonic on that disk; connectedness propagates finiteness to the whole surface.
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