Editing Almost all (section)

===Meaning in measure theory===
{{further|Almost everywhere}}
[[File:CantorEscalier.svg|thumb|right|250px| The [[Cantor function]] as a function that has zero derivative almost everywhere]]

When speaking about the [[real number|reals]], sometimes "almost all" can mean "all reals except for a [[null set]]".{{r|Korevaar|Natanson}}{{r|Clapham|group=sec}} Similarly, if <var>S</var> is some set of reals, "almost all numbers in <var>S</var>" can mean "all numbers in <var>S</var> except for those in a null set".{{r|Sohrab}} The [[real line]] can be thought of as a one-dimensional [[Euclidean space]]. In the more general case of an <var>n</var>-dimensional space (where <var>n</var> is a positive integer), these definitions can be [[generalised]] to "all points except for those in a null set"{{r|James|group=sec}} or "all points in <var>S</var> except for those in a null set" (this time, <var>S</var> is a set of points in the space).{{r|Helmberg}} Even more generally, "almost all" is sometimes used in the sense of "[[almost everywhere]]" in [[measure theory]],{{r|Vestrup|Billingsley}}{{r|Bityutskov|group=sec}} or in the closely related sense of "[[almost surely]]" in [[probability theory]].{{r|Billingsley}}{{r|Ito2|group=sec}}

Examples:
* In a [[measure space]], such as the real line, countable sets are null. The set of [[rational number]]s is countable, so almost all real numbers are irrational.{{r|Niven}}
* Georg [[Cantor's first set theory article]] proved that the set of [[algebraic number]]s is countable as well, so almost all reals are [[transcendental number|transcendental]].{{r|Baker}}{{r|group=sec|RealTrans}}
* Almost all reals are [[normal number|normal]].{{r|Granville}}
* The [[Cantor set]] is also null. Thus, almost all reals are not in it even though it is uncountable.{{r|Korevaar}}
* The derivative of the [[Cantor function]] is 0 for almost all numbers in the [[unit interval]].{{r|Burk}} It follows from the previous example because the Cantor function is [[locally constant function|locally constant]], and thus has derivative 0 outside the Cantor set.