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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 S = { n ∈ ∈ N | n ≥ ≥ k } {\displaystyle S=\{n\in \mathbb {N} \,|\,n\geq k\}} is almost N {\displaystyle \mathbb {N} } for any k {\displaystyle k} in N {\displaystyle \mathbb {N} } , because only finitely many natural numbers are less than k {\displaystyle k} .

• The set of prime numbers is not almost N {\displaystyle \mathbb {N} } , because there are infinitely many natural numbers that are not prime numbers.

• The set of transcendental numbers are almost R {\displaystyle \mathbb {R} } , because the algebraic real numbers form a countable subset of the set of real numbers (which is uncountable).cite-ref-1[1]

• The Cantor set is uncountably infinite, but has Lebesgue measure zero.cite-ref-2[2] So almost all real numbers in (0, 1) are members of the complement of the Cantor set.

• See also

• References

Look up

almost

in Wiktionary, the free dictionary.

• Almost periodic function - and Operators

• Almost all

• Almost surely

• Approximation

• List of mathematical jargon

cite-note-11. ↑ "Almost All Real Numbers are Transcendental - ProofWiki". proofwiki.org. Retrieved 2019-11-16.

cite-note-22. ↑ "Theorem 36: the Cantor set is an uncountable set with zero measure". Theorem of the week. 2010-09-30. Retrieved 2019-11-16.