Definition:Almost All/Set Theory/Uncountable

Definition

Let $S$ be an uncountably infinite set.

Let $P: S \to \set {\text {true}, \text {false} }$ be a property of $S$ such that:

$\set {s \in S: \neg \map P s}$

is countable (either finite or countably infinite).

Then $P$ holds for almost all of the elements of $S$.