PomeDefinition 3.4.2.3E4C6A9CCF19 A function $u\colon \Omega \to [-\infty ,\infty )$ is subharmonic if it is upper semicontinuous and every $x\in \Omega $ admits an $r_ x>0$ with $\overline{B(x,r_ x)}\subset \Omega $ such that
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3.4.3
\begin{equation} u(x)\leq \frac{1}{s_{d-1}}\int _{S^{d-1}}u(x+r\omega )\, \mathrm d S(\omega )\qquad \text{for }0<r<r_ x. \end{equation}
The function identically equal to $-\infty $ is subharmonic. We write $\mathrm{SH}(\Omega )$ for the set of subharmonic functions on $\Omega $. A function $v$ is superharmonic if $-v$ is subharmonic.
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