Subharmonic functions on plane domains

Let $\mathrm{\Omega }\subseteq \mathbb{C}$ be a complex domain. A function $u:\mathrm{\Omega }\to [-\mathrm{\infty },\mathrm{\infty })$ is subharmonic on $\mathrm{\Omega }$ when:

1. $u$ is upper semicontinuous;
2. on no connected component of $\mathrm{\Omega }$ is $u$ identically $-\mathrm{\infty }$;
3. for every closed disc $\overline{D(a,r)}\subseteq \mathrm{\Omega }$, $u(a)\le \frac{1}{2\pi }{\int }_0^{2\pi }u(a+r{e}^{it})dt,$ where the integral is taken in the extended-real sense.

The radius $r$ is always positive. The value $u(a)$ is allowed to be $-\mathrm{\infty }$, in which case the submean inequality is automatic.

On a circle, upper semicontinuity gives a finite upper bound, so the integral above can only fail in the downward direction; the next lemma records that the boundary function is Borel and that the average is therefore defined in $[-\mathrm{\infty },\mathrm{\infty })$.