- 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" (formally, a

cofinite set) or "all but acountable set " (formally, acocountable set); seealmost . An example of this usage is the "Frivolous Theorem of Arithmetic", which states that "almost all natural numbers are very, very, very large". [*MathWorld|urlname=FrivolousTheoremofArithmetic|title=Frivolous Theorem of Arithmetic*]When speaking about the reals, sometimes it means "all reals but a set of

Lebesgue measure zero" (formally,almost everywhere ). In this sense we can say "almost all reals are not a member of theCantor set ".In

number theory , if "P"("n") is a property of positiveinteger s, and if "p"("N") denotes the number ofpositive 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" (formally,

asymptotically almost surely ) and write:$(forall^infty\; n)\; P(n).$For example, the

prime number theorem states that the number ofprime 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 (not prime), however there are still an infinite number of primes.Occasionally, "almost all" is used in the sense of "

almost everywhere " inmeasure theory , or in the closely related sense of "almost surely " inprobability theory .**ee also***

Sufficiently large **References**

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