proof of capacity generated by a measure
For a finite measure space , define
We show that is an -capacity and that a subset is -capacitable (http://planetmath.org/ChoquetCapacity) if and only if it is in the completion (http://planetmath.org/CompleteMeasure) of with respect to .
Note, first of all, that for any . That is increasing follows directly from the definition. If is a decreasing sequence of sets then is also in and, by continuity from above (http://planetmath.org/PropertiesForMeasure) for measures,
Now suppose that is an increasing sequence of subsets of and set . Then, for each and, hence, .
Additivity of then gives
So, by continuity from below for measures,
Choosing arbitrarily small shows that and, therefore, is indeed an -capacity.
Now suppose that is in the completion of with respect to , so that there exists with and . Then,
and is indeed -capacitable. Conversely, let be -capacitable. Then, there exists such that and
Setting and gives and
So , as required.
|Title||proof of capacity generated by a measure|
|Date of creation||2013-03-22 18:47:55|
|Last modified on||2013-03-22 18:47:55|
|Last modified by||gel (22282)|