= Harmonic lifting of a continuous subharmonic function
{title2=$L_Vu$}
On a relatively compact coordinate disk $V$ in a <Riemann surface>, replace a continuous <subharmonic function> $u$ inside $V$ by the <Poisson integral> of its boundary values, leaving it unchanged outside. The resulting function $L_Vu$ is continuous and subharmonic, equals $u$ off $V$, is harmonic on $V$, and satisfies $L_Vu\geq u$. The <maximum principle for subharmonic functions> proves the inequality. The pasting assertion follows from harmonic comparison: a harmonic comparison function dominating the pasted function on the boundary of a test domain first dominates $u$, then dominates the disk lift on its part inside $V$.
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