Superharmonic function

Remarks

A function $u$ is superharmonic if and only if $-u$ is subharmonic and not identically $-\infty$ on any component. It suffices to require clause 3 for all small enough $r$ at each point. A function $u$ of class $C^2$ is superharmonic if and only if its Laplacian satisfies $\Delta u \le 0$. A function is harmonic if and only if it is both superharmonic and subharmonic.

Example: for $n \ge 3$ let $u(x) = \abs{x}^{2-n}$ on $\R^n$, with $u(0) = +\infty$. It is lower semicontinuous and finite off $0$. Its mean over $S(a, r)$ equals $\max(\abs{a}, r)^{2-n}$, which is at most $\abs{a}^{2-n} = u(a)$, and at $a = 0$ the value $+\infty$ dominates. So $u$ is superharmonic. In the plane, $u(x) = -\log\abs{x}$ is superharmonic in the same way.

Non-example: $u(x) = \abs{x}^2$ is continuous on $\R^n$, but its mean over $S(a, r)$ is $\abs{a}^2 + r^2 > u(a)$, so clause 3 fails.