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 <math> S = \{ n \in \mathbb{N}\,|\, n \ge k \} </math> is almost <math>\mathbb{N}</math> for any <math>k</math> in <math>\mathbb{N}</math>, because only finitely many natural numbers are less than <math>k</math>.
- The set of prime numbers is not almost <math>\mathbb{N}</math>, because there are infinitely many natural numbers that are not prime numbers.
- The set of transcendental numbers are almost <math>\mathbb{R}</math>, because the algebraic real numbers form a countable subset of the set of real numbers (which is uncountable).
- The Cantor set is uncountably infinite, but has Lebesgue measure zero. So almost all real numbers in (0, 1) are members of the complement of the Cantor set.
See also
- Almost periodic function - and Operators
- Almost all
- Almost surely
- Approximation
- List of mathematical jargon