capacity generated by a measure

Any finite measureMathworldPlanetmath ( can be extended to a set functionMathworldPlanetmath on the power setMathworldPlanetmath of the underlying space. As the following result states, this will be a Choquet capacity.


Let (X,F,μ) be a finite measure space. Then,


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

Note that, as well as being a capacity, μ* is also an outer measureMathworldPlanetmathPlanetmath (see here (, which does not require the finiteness of μ. Clearly, μ*(A)=μ(A) for all A, so μ* is an extensionPlanetmathPlanetmath of μ to the power set of X, and is referred to as the outer measure generated by μ.

Recall that a subset SX is in the completion of with respect to μ if and only if there are sets A,B with ASB and μ(BA)=0 which, by the above theorem, is equivalentMathworldPlanetmathPlanetmathPlanetmathPlanetmathPlanetmath to the capacitability of S.

Title capacity generated by a measure
Canonical name CapacityGeneratedByAMeasure
Date of creation 2013-03-22 18:47:35
Last modified on 2013-03-22 18:47:35
Owner gel (22282)
Last modified by gel (22282)
Numerical id 7
Author gel (22282)
Entry type Theorem
Classification msc 28A05
Classification msc 28A12
Synonym outer measure generated by a measure
Defines outer measure generated by