completion of a measure space
If the measure space is not complete, then it can be completed in the following way. Let
i.e. the family of all subsets of sets whose -measure is zero. Define
We assert that is a -algebra. In fact, it clearly contains the emptyset, and it is closed under countable unions because both and are. We thus need to show that it is closed under complements. Let , and suppose is such that and . Then we have
where and . Hence .
Now we define on by , whenever and . It is easily verified that this defines in fact a measure, and that is the completion of .
|Title||completion of a measure space|
|Date of creation||2013-03-22 14:06:59|
|Last modified on||2013-03-22 14:06:59|
|Last modified by||Koro (127)|