# Almost

In set theory, when dealing with sets of infinite size, the term almost or nearly is used to refer to all but a negligible amount of elements in the set. The notion of "negligible" depends on the context, and may mean "of measure zero" (in a measure space), "finite" (when infinite sets are involved), or "countable" (when uncountably infinite sets are involved).

For example:

• The set $\displaystyle{ S = \{ n \in \mathbb{N}\,|\, n \ge k \} }$ is almost $\displaystyle{ \mathbb{N} }$ for any $\displaystyle{ k }$ in $\displaystyle{ \mathbb{N} }$, because only finitely many natural numbers are less than $\displaystyle{ k }$.
• The set of prime numbers is not almost $\displaystyle{ \mathbb{N} }$, because there are infinitely many natural numbers that are not prime numbers.
• The set of transcendental numbers are almost $\displaystyle{ \mathbb{R} }$, because the algebraic real numbers form a countable subset of the set of real numbers (which is uncountable).[1]
• The Cantor set is uncountably infinite, but has Lebesgue measure zero.[2] So almost all real numbers in (0, 1) are members of the complement of the Cantor set.