Almost

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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]\displaystyle{ S = \{ n \in \mathbb{N}\,|\, n \ge k \} }[/math] is almost [math]\displaystyle{ \mathbb{N} }[/math] for any [math]\displaystyle{ k }[/math] in [math]\displaystyle{ \mathbb{N} }[/math], because only finitely many natural numbers are less than [math]\displaystyle{ k }[/math].
  • The set of prime numbers is not almost [math]\displaystyle{ \mathbb{N} }[/math], because there are infinitely many natural numbers that are not prime numbers.
  • The set of transcendental numbers are almost [math]\displaystyle{ \mathbb{R} }[/math], 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.

See also

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