proof of extending a capacity to a Cartesian product
Let (X,ℱ) be a paved space such that ℱ is closed under finite unions and finite intersections
, and (K,𝒦) be a compact
paved space.
Define 𝒢 to be the closure
under finite unions and finite intersections of the paving ℱ×𝒦 on X×K.
For an ℱ-capacity I, define
˜I:𝒫(X×K)→ℝ, | ||
˜I(S)=I(πX(S)), |
where πX is the projection map onto X. We show that ˜I is a 𝒢-capacity and that πX(S)∈ℱδ whenever S∈𝒢δ.
Clearly, the property that ˜I is an increasing set function follows from the fact that I satisfies this property. Furthermore, if Sn⊆X×K is an increasing sequence of sets with S=⋃nSn then πX(Sn) is an increasing sequence and
˜I(S)=I(πX(S))=I(⋃nπX(Sn))=lim |
To prove that is a -capacity, it only remains to show that if is a sequence in decreasing to then .
Note that any in can be written as for sets and . The projection onto is then
which, as is closed under finite unions and finite intersections, must be in . Furthermore, for any ,
This shows that is in the closure of under finite unions and finite intersections. Furthermore, since compact pavings are closed subsets of a compact topological space (http://planetmath.org/CompactPavingsAreClosedSubsetsOfACompactSpace), is itself a compact paving.
Now let be a decreasing sequence of sets in and set . Then for each , giving . To prove the reverse inequality, consider . Then, is a nonempty set in for all . By compactness, must also be nonempty and therefore . This shows that
Furthermore, as we have shown that and, as is an -capacity,
So is a -capacity.
We finally show that if then . By definition, there is a sequence such that . Setting then, since is closed under finite unions and finite intersections, . Furthermore, decreases to so, as shown above, and
as required.
Title | proof of extending a capacity to a Cartesian product |
---|---|
Canonical name | ProofOfExtendingACapacityToACartesianProduct |
Date of creation | 2013-03-22 18:47:41 |
Last modified on | 2013-03-22 18:47:41 |
Owner | gel (22282) |
Last modified by | gel (22282) |
Numerical id | 5 |
Author | gel (22282) |
Entry type | Proof |
Classification | msc 28A12 |
Classification | msc 28A05 |