Quasi-everywhere

Remarks

Example: let $K$ be the closed unit disc in $\C$ and let $u(z) = \log \abs{z}$. Then $u > -\infty$ quasi-everywhere on $K$, since the exceptional set $\set{0}$ is finite and so has capacity zero.

Non-example: let $u$ be the function on $\C$ with $u = 1$ on the segment $[0, 1]$ and $u = 0$ elsewhere. Then $u = 0$ almost everywhere with respect to area, but not quasi-everywhere, because the segment has positive logarithmic capacity.

In $\R^n$ with $n \ge 3$ the capacity used is the Newtonian capacity, and in the plane it is the logarithmic capacity. In nonlinear potential theory one speaks of $p$-quasi-everywhere, with the $p$-capacity in place of the Newtonian capacity.