Editing Almost everywhere (section)

== Properties ==

* If property <math>P </math> holds almost everywhere and implies property ''<math>Q </math>'', then property ''<math>Q </math>'' holds almost everywhere. This follows from the [[Measure (mathematics)#Monotonicity|monotonicity]] of measures.
* If <math> (P_n)  </math> is a finite or a countable sequence of properties, each of which holds almost everywhere, then their conjunction <math> \forall n P_n  </math> holds almost everywhere. This follows from the [[Measure (mathematics)#Measures of infinite unions of measurable sets|countable sub-additivity]] of measures.
* By contrast, if <math> (P_x)_{x\in \mathbf R}  </math> is an uncountable family of properties, each of which holds almost everywhere, then their conjunction <math> \forall x P_x  </math> does not necessarily hold almost everywhere. For example, if <math>\mu </math> is Lebesgue measure on <math>X = \mathbf R </math> and <math> P_x  </math> is the property of not being equal to <math> x  </math> (i.e. <math> P_x(y)  </math> is true if and only if <math> y \neq x </math>), then each <math> P_x  </math> holds almost everywhere, but the conjunction <math> \forall x P_x  </math> does not hold anywhere.

As a consequence of the first two properties, it is often possible to reason about "almost every point" of a measure space as though it were an ordinary point rather than an abstraction.{{cn|reason=Give an example of this kind of reasoning.|date=April 2019}} This is often done implicitly in informal mathematical arguments. However, one must be careful with this mode of reasoning because of the third bullet above: universal quantification over uncountable families of statements is valid for ordinary points but not for "almost every point".