Almost all
In mathematics, the phrase almost all has a number of specialised uses.
"Almost all" is sometimes used synonymously with "all but finitely many" or "all but a countable set"; see almost. An example of this usage is the Frivolous Theorem of Arithmetic, which states that almost all natural numbers are very, very, very large.[1]
When speaking about the reals, sometimes it means "all reals but a set of Lebesgue measure zero". In this sense we can say "almost all reals are irrational".
In number theory, if a subset A of the natural numbers has natural density 1, then we say that almost all (natural) numbers belong to A. Let us speak of properties instead of sets and apply the definition of natural density. 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" and write
- .
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, however there are still an infinite number of primes.
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.