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Almost all

In mathematics, the phrase almost all has three specialised uses:

  1. "Almost all" is sometimes used synonymous with "all but finitely many"; see almost.
  2. 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 tends to 1 as N tends to ∞ (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.
  3. Occasionally, "almost all" is used in the sense of "almost everywhere" in measure theory.