Almost

Short description: Term in set theory

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 $S=\{n\in \mathbb{\mathbb{N}}|n\ge k\}$ is almost $\mathbb{\mathbb{N}}$ for any $k$ in $\mathbb{\mathbb{N}}$, because only finitely many natural numbers are less than $k$.
• The set of prime numbers is not almost $\mathbb{\mathbb{N}}$, 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 numbers (which is uncountable).
• The Cantor set is uncountably infinite, but has Lebesgue measure zero. So almost all real numbers in (0, 1) are members of the complement of the Cantor set.

See also

• Almost periodic function - and Operators
• Almost all
• Almost surely
• Approximation
• List of mathematical jargon

References

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