extending a capacity to a Cartesian product

A capacity on a set X can be extended to a set function on a Cartesian product X×K simply by projecting any subset onto X, and then applying the original capacity.


Suppose that (X,F) is a paved space such that F is closed under finite unions and finite intersectionsMathworldPlanetmath, and that (K,K) is a compact paved space. Define G to be the closure under finite unions and finite intersections of the paving F×K on X×K.

If I is an F-capacity and πX:X×KX is the projection map, we can form the compositionMathworldPlanetmathPlanetmath


Then πX(S)Fδ for any SGδ, and IπX is a G-capacity.

This result justifies looking at capacities when considering projections from the Cartesian product X×K onto X. We see that the property of being a capacity is preserved under composing with such projections. However, additivity of set functions is not preserved, so the corresponding result would not be true if “capacity” was replaced by “measureMathworldPlanetmath” or “outer measureMathworldPlanetmathPlanetmath”.

Recall that if SX×K is (𝒢,IπX)-capacitable then, for any ϵ>0, there is an A𝒢δ such that AS and IπX(A)>IπX(S)-ϵ. However, πX(A)πX(S) and, by the above theorem, πX(A)δ. This has the following consequence.


Let SX×K be (G,IπX)-capacitable. Then, πX(S) is (F,I)-capacitable.

Title extending a capacity to a Cartesian product
Canonical name ExtendingACapacityToACartesianProduct
Date of creation 2013-03-22 18:47:38
Last modified on 2013-03-22 18:47:38
Owner gel (22282)
Last modified by gel (22282)
Numerical id 6
Author gel (22282)
Entry type Theorem
Classification msc 28A12
Classification msc 28A05