Editing Almost everywhere (section)

== Definition ==

If <math>(X,\Sigma,\mu) </math> is a [[measure space]], a property <math>P </math> is said to hold almost everywhere in <math>X </math> if there exists a measurable set <math>N \in \Sigma </math> with <math>\mu(N) = 0 </math>, and all <math>x\in X\setminus N </math> have the property <math>P </math>.<ref>{{Cite web|url=http://mathonline.wikidot.com/properties-that-hold-almost-everywhere|title=Properties That Hold Almost Everywhere - Mathonline|website=mathonline.wikidot.com|access-date=2019-11-19}}</ref> 
Another common way of expressing the same thing is to say that "almost every point satisfies <math>P\, </math>", or that "for almost every <math>x </math>, <math>P(x) </math> holds".

It is ''not'' required that the set <math>\{x\in X: \neg P(x)\} </math> has measure zero; it may not be measurable. 
By the above definition, it is sufficient that  <math>\{x\in X: \neg P(x)\} </math> be contained in some set <math>N </math> that is measurable and has measure zero. 
However, this technicality vanishes when considering a [[complete measure space]]: if <math>X</math> is complete then <math>N</math> exists with measure zero if and only if <math>\{x\in X: \neg P(x)\}</math> is measurable with measure zero.