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