In mathematics, the phrase almost all has a number of specialised uses.
"Almost all" is sometimes used synonymously with "all but finitely many"; see almost.
In number theory, if P(n) is a property of positive integers, and if p(N) denotes the number of positive integers n less than N for which P(n) holds, and if
- p(N)/N → 1 as N → ∞
(see limit), then we say that "P(n) holds for almost all positive integers n". For example, the prime number theorem states that the number of prime numbers less than or equal to N is asymptotically equal to N/ln N. Therefore the proportion of prime integers is roughly 1/ln N, which tends to 0. Thus, almost all positive integers are composite.
Occasionally, "almost all" is used in the sense of "almost everywhere" in measure theory, or in the closely related sense of "almost surely" in probability theory.
See also