Editing Almost everywhere (section)

{{Short description|Everywhere except a set of measure zero}}
[[File:Function-1 x.svg|thumb|The function [[1/x]] is [[differentiable]] and [[Continuous function|continuous]] almost everywhere, more precisely, everywhere except at x = 0.]]

In [[measure theory]] (a branch of [[mathematical analysis]]), a property holds '''almost everywhere''' if, in a technical sense, the set for which the property holds takes up nearly all possibilities. The notion of "almost everywhere" is a companion notion to the concept of [[measure zero]], and is analogous to the notion of ''[[almost surely]]'' in [[probability theory]].

More specifically, a property holds almost everywhere if it holds for all elements in a set except a subset of measure zero,<ref>{{Cite web|url=http://mathworld.wolfram.com/AlmostEverywhere.html|title=Almost Everywhere|last=Weisstein|first=Eric W.|website=mathworld.wolfram.com|language=en|access-date=2019-11-19}}</ref><ref>{{Cite book|url=https://archive.org/details/measuretheory00halm|title=Measure theory|last=Halmos|first=Paul R.|publisher=Springer-Verlag|year=1974|isbn=0-387-90088-8|location=New York}}</ref> or equivalently, if the set of elements for which the property holds is [[conull set|conull]]. In cases where the measure is not [[Complete measure|complete]], it is sufficient that the set be contained within a set of measure zero. When discussing sets of [[real number]]s, the [[Lebesgue measure]] is usually assumed unless otherwise stated.

The term ''almost everywhere'' is abbreviated ''a.e.'';<ref>{{Cite web|url=https://www.dictionary.com/browse/almost-everywhere|title=Definition of almost everywhere {{!}} Dictionary.com|website=www.dictionary.com|language=en|access-date=2019-11-19}}</ref> in older literature ''p.p.'' is used, to stand for the equivalent [[French language]] phrase ''presque partout''.<ref>{{Cite journal|last=Ursell|first=H. D.|date=1932-01-01|title=On the Convergence Almost Everywhere of Rademacher's Series and of the Bochnerfejér Sums of a Function almost Periodic in the Sense of Stepanoff|url=https://academic.oup.com/plms/article/s2-33/1/457/1523180|journal=Proceedings of the London Mathematical Society|language=en|volume=s2-33|issue=1|pages=457–466|doi=10.1112/plms/s2-33.1.457|issn=0024-6115}}</ref>

A set with '''full measure''' is one whose complement is of measure zero. In probability theory, the terms ''almost surely'', ''almost certain'' and ''almost always'' refer to [[event (probability theory)|event]]s with [[probability]] 1 not necessarily including all of the outcomes. These are exactly the sets of full measure in a probability space.

Occasionally, instead of saying that a property holds almost everywhere, it is said that the property holds for '''almost all''' elements (though the term [[almost all]] can also have other meanings).