Editing Complete measure

{{Short description|Measure space where every subset of a set with null measure is measurable (and has null measure)}}
{{Refimprove|date=October 2010}}
In [[mathematics]], a '''complete measure''' (or, more precisely, a '''complete measure space''') is a [[measure (mathematics)|measure space]] in which every [[subset]] of every [[null set]] is measurable (having [[measure zero]]). More formally, a measure space (''X'',&nbsp;Σ,&nbsp;''μ'') is complete if and only if<ref>{{Cite book |last=Halmos |first=Paul R. |url=http://link.springer.com/10.1007/978-1-4684-9440-2 |title=Measure Theory |date=1950 |publisher=Springer New York |isbn=978-1-4684-9442-6 |series=Graduate Texts in Mathematics |volume=18 |location=New York, NY |pages=31 |doi=10.1007/978-1-4684-9440-2}}</ref><ref>{{Cite book |last=de Barra |first=G. |url=http://www.sciencedirect.com/science/book/9781904275046 |title=Measure theory and integration |date=2003 |publisher=Woodhead Publishing Limited |isbn=978-1-904275-04-6 |pages=94 |doi=10.1533/9780857099525}}</ref>

:<math>S \subseteq N \in \Sigma \mbox{ and } \mu(N) = 0\ \Rightarrow\ S \in \Sigma.</math>

==Motivation==
The need to consider questions of completeness can be illustrated by considering the problem of product spaces.

Suppose that we have already constructed [[Lebesgue measure]] on the [[real line]]: denote this measure space by <math>(\R, B, \lambda).</math> We now wish to construct some two-dimensional Lebesgue measure <math>\lambda^2</math> on the plane <math>\R^2</math> as a [[product measure]]. Naively, we would take the [[Sigma algebra|{{sigma}}-algebra]] on <math>\R^2</math> to be <math>B \otimes B,</math> the smallest {{sigma}}-algebra containing all measurable "rectangles" <math>A_1 \times A_2</math> for <math>A_1, A_2 \in B.</math>

While this approach does define a [[measure space]], it has a flaw. Since every [[Singleton (mathematics)|singleton]] set has one-dimensional Lebesgue measure zero,
<math display=block>\lambda^2(\{0\} \times A) \leq \lambda(\{0\}) = 0</math>
for {{em|any}} subset <math>A</math> of <math>\R.</math> However, suppose that <math>A</math> is a [[Non-measurable set|non-measurable subset]] of the real line, such as the [[Vitali set]]. Then the <math>\lambda^2</math>-measure of <math>\{0\} \times A</math> is not defined but
<math display=block>\{0\} \times A \subseteq \{0\} \times \R,</math>
and this larger set does have <math>\lambda^2</math>-measure zero. So this "two-dimensional Lebesgue measure" as just defined is not complete, and some kind of completion procedure is required.

==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>

==Examples==
* [[Borel measure]] as defined on the Borel ''σ''-algebra generated by the [[Open set|open]] [[Interval (mathematics)|intervals]] of the real line is not complete, and so the above completion procedure must be used to define the complete Lebesgue measure. This is illustrated by the fact that the set of all Borel sets over the reals has the same cardinality as the reals. While the [[Cantor set]] is a Borel set, has measure zero, and its power set has cardinality strictly greater than that of the reals. Thus there is a subset of the Cantor set that is not contained in the Borel sets. Hence, the Borel measure is not complete.
* ''n''-dimensional Lebesgue measure is the completion of the ''n''-fold product of the one-dimensional Lebesgue space with itself. It is also the completion of the Borel measure, as in the one-dimensional case.

==Properties==

[[Maharam's theorem]] states that every complete measure space is decomposable into measures on [[Continuum (set theory)|continua]], and a finite or countable [[counting measure]].

==See also==

* {{annotated link|Inner measure}}
* {{annotated link|Lebesgue measurable set}}

==References==
* {{SpringerEOM |title=Complete measure |id=C/c023800 |first=A.P. |last=Terekhin}}
{{DEFAULTSORT:Complete Measure}}
<references />{{Measure theory}}

[[Category:Measures (measure theory)]]