Editing Complete measure (section)

==Construction of a complete measure==
Given a (possibly incomplete) measure space (''X'',&nbsp;Σ,&nbsp;''μ''), there is an extension (''X'',&nbsp;Σ<sub>0</sub>,&nbsp;''μ''<sub>0</sub>) of this measure space that is complete.<ref>{{Cite book |last=Rudin |first=Walter |title=Real and complex analysis |date=2013 |publisher=McGraw-Hill |isbn=978-0-07-054234-1 |edition=3. ed., internat. ed., [Nachdr.] |series=McGraw-Hill international editions Mathematics series |location=New York, NY |pages=27–28}}</ref> The smallest such extension (i.e. the smallest ''σ''-algebra Σ<sub>0</sub>) is called the '''completion''' of the measure space.

The completion can be constructed as follows:
* let ''Z'' be the set of all the subsets of the zero-''μ''-measure subsets of ''X'' (intuitively, those elements of ''Z'' that are not already in Σ are the ones preventing completeness from holding true);
* let Σ<sub>0</sub> be the ''σ''-algebra generated by Σ and ''Z'' (i.e. the smallest ''σ''-algebra that contains every element of Σ and of ''Z'');
* ''μ'' has an extension ''μ''<sub>0</sub> to Σ<sub>0</sub> (which is unique if ''μ'' is [[Σ-finite measure|''σ''-finite]]), called the [[outer measure]] of ''μ'', given by the [[infimum]]

::<math>\mu_{0} (C) := \inf \{ \mu (D) \mid C \subseteq D \in \Sigma \}.</math>

Then (''X'',&nbsp;Σ<sub>0</sub>,&nbsp;''μ''<sub>0</sub>) is a complete measure space, and is the completion of (''X'',&nbsp;Σ,&nbsp;''μ'').

In the above construction it can be shown that every member of Σ<sub>0</sub> is of the form ''A''&nbsp;∪&nbsp;''B'' for some ''A''&nbsp;∈&nbsp;Σ and some ''B''&nbsp;∈&nbsp;''Z'', and

:<math>\mu_{0} (A \cup B) = \mu (A).</math>