Choquet capacity

A Choquet capacity, or just capacity, on a set X is a kind of set functionMathworldPlanetmath, mapping the power setMathworldPlanetmath 𝒫(X) to the real numbers.


Let F be a collectionMathworldPlanetmath of subsets of X. Then, an F-capacity is an increasing set function


satisfying the following.

  1. 1.

    If (An)n is an increasing sequence of subsets of X then I(An)I(mAm) as n.

  2. 2.

    If (An)n is a decreasing sequence of subsets of X such that An for each n, then I(An)I(mAm) as n.

The condition that I is increasing means that I(A)I(B) whenever AB. Note that capacities differ from the concepts of measuresMathworldPlanetmath and outer measuresMathworldPlanetmathPlanetmath, as no additivity or subadditivity conditions are imposed. However, for any finite measure, there is a corresponding capacity ( An important application to the theory of measures and analytic setsMathworldPlanetmath is given by the capacitability theorem.

The (F,I)-capacitable sets are defined as follows. Recall that δ denotes the collection of countableMathworldPlanetmath intersections of sets in the paving .


Let I be an F-capacity on a set X. Then a subset AX is (,I)-capacitable if, for each ϵ>0, there exists a BFδ such that BA and I(B)I(A)-ϵ.

Alternatively, such sets are called I-capacitable or, simply, capacitable.

Title Choquet capacity
Canonical name ChoquetCapacity
Date of creation 2013-03-22 18:47:26
Last modified on 2013-03-22 18:47:26
Owner gel (22282)
Last modified by gel (22282)
Numerical id 5
Author gel (22282)
Entry type Definition
Classification msc 28A12
Classification msc 28A05
Synonym capacity
Defines capacitable