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 ${\displaystyle S=\{n\in \mathbb {N} \,|\,n\geq k\}}$ 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.

See also

Look up almost in Wiktionary, the free dictionary.

1. ^ "Almost All Real Numbers are Transcendental - ProofWiki". proofwiki.org. Retrieved 2019-11-16.
2. ^ "Theorem 36: the Cantor set is an uncountable set with zero measure". Theorem of the week. 2010-09-30. Retrieved 2019-11-16.