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 in the context, and may mean "of measure zero" (in a measure space), "countable" (when uncountably infinite sets are involved), or "finite" (when infinite sets are involved).^[1]

For example:

The set $S=\{n\in \mathbb {N} \,|\,n\geq k\}$ is almost $\mathbb {N}$ for any $k$ in $\mathbb {N}$ , because only finitely many natural numbers are less than $k$ .
The set of prime numbers is not almost $\mathbb {N}$ , because there are infinitely many natural numbers that are not prime numbers.
The set of transcendental numbers are almost $\mathbb {R}$ , because the algebraic real numbers form a countable subset of the set of real number (the latter of which is uncountable).^[2]
The Cantor set is uncountably infinite, but has Lebesgue measure zero.^[3] So almost all real numbers in (0, 1) are member of the complement of the Cantor set.

References

^ "The Definitive Glossary of Higher Mathematical Jargon — Almost". Math Vault. 2019-08-01. Retrieved 2019-11-16.
^ "Almost All Real Numbers are Transcendental - ProofWiki". proofwiki.org. Retrieved 2019-11-16.
^ "Theorem 36: the Cantor set is an uncountable set with zero measure". Theorem of the week. 2010-09-30. Retrieved 2019-11-16.

[1] "The Definitive Glossary of Higher Mathematical Jargon — Almost". Math Vault. 2019-08-01. Retrieved 2019-11-16.

[2] "Almost All Real Numbers are Transcendental - ProofWiki". proofwiki.org. Retrieved 2019-11-16.

[3] "Theorem 36: the Cantor set is an uncountable set with zero measure". Theorem of the week. 2010-09-30. Retrieved 2019-11-16.

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Almost

See also

References