---
tag: 1X1WCNDA0J
kind: concept
name: Quasi-everywhere
msc: [31B15, 31A15, 31C15]
sources:
  - url: https://en.wikipedia.org/wiki/Capacity_of_a_set
---

## Summary

A property holds quasi-everywhere on a set if it holds at every point outside a set of [[capacity zero|W9N1TR1X8P]].
It plays the role in potential theory that "almost everywhere" plays in measure theory, with capacity in place of measure.
Since sets of capacity zero have Lebesgue measure zero, a property that holds quasi-everywhere also holds almost everywhere.
Many results, such as [[Frostman's theorem|TX0Z2VVKH3]] and the solvability of the Dirichlet problem at boundary points, hold quasi-everywhere but not everywhere.

## Explanations

### L1 human

"Almost everywhere" allows exceptions on a set of zero volume, which can still be quite large, like a whole line in the plane.
"Quasi-everywhere" allows only exceptions on a set of capacity zero, which is much thinner.
For example, the potential of the equilibrium charge on a conductor is constant at every point of the conductor except on a set of capacity zero.
So quasi-everywhere statements are sharper than almost-everywhere statements.