# Almost

From HandWiki

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

## References

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Original source: https://en.wikipedia.org/wiki/Almost.
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- ↑ "Almost All Real Numbers are Transcendental - ProofWiki". https://proofwiki.org/wiki/Almost_All_Real_Numbers_are_Transcendental.
- ↑ "Theorem 36: the Cantor set is an uncountable set with zero measure" (in en). 2010-09-30. https://theoremoftheweek.wordpress.com/2010/09/30/theorem-36-the-cantor-set-is-an-uncountable-set-with-zero-measure/.