Almost

For other uses, see Almost (disambiguation).

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 is almost for any in , because only finitely many natural numbers are less than .
  • The set of prime numbers is not almost , because there are infinitely many natural numbers that are not prime numbers.
  • The set of transcendental numbers are almost , 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.

References

  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.


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