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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