Completion


Definition of completion
In a measure space , all subsets satisfying belong to .

Completion of measure space .
Let's write

(symmetric difference of and ).
For , if satisfying

, then the entire is defined as .
is a additive class.
Let's consider satisfying the above conditions with respect to and define .
The measure space is complete, and it is also the smallest extension.

Outer measure completion
In measure space , let's define
.
is an outer measure.
is equal to the completion of

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