proof of capacity generated by a measure

For a finite measure space (X,,μ), define


We show that μ* is an -capacity and that a subset SX is (,μ*)-capacitable ( if and only if it is in the completionPlanetmathPlanetmath ( of with respect to μ.

Note, first of all, that μ*(S)=μ(S) for any S. That μ* is increasing follows directly from the definition. If An is a decreasing sequence of sets then A=nAn is also in and, by continuity from above ( for measuresMathworldPlanetmath,


as n.

Now suppose that Sn is an increasing sequence of subsets of X and set S=nSn. Then, μ*(Sn)μ*(S) for each n and, hence, limnμ*(Sn)μ*(S).

To prove the reverse inequalityMathworldPlanetmath, choose any ϵ>0 and sequenceMathworldPlanetmath An with SnAn and μ(An)μ*(Sn)+2-nϵ. Then, AmAnSn whenever mn and, therefore,


Additivity of μ then gives


So, by continuity from below for measures,


Choosing ϵ arbitrarily small shows that μ*(Sn)μ*(S) and, therefore, μ* is indeed an -capacity.

Now suppose that S is in the completion of with respect to μ, so that there exists A,B with ASB and μ(BA)=0. Then,


and S is indeed (,μ*)-capacitable. Conversely, let S be (,μ*)-capacitable. Then, there exists An,Bn such that AnSBn and


Setting A=nAn and B=nBn gives ASB and


So μ(BA)=μ(B)-μ(A)=0, as required.

Title proof of capacity generated by a measure
Canonical name ProofOfCapacityGeneratedByAMeasure
Date of creation 2013-03-22 18:47:55
Last modified on 2013-03-22 18:47:55
Owner gel (22282)
Last modified by gel (22282)
Numerical id 5
Author gel (22282)
Entry type Proof
Classification msc 28A12
Classification msc 28A05