proof of extending a capacity to a Cartesian product
Let be a paved space such that is closed under finite unions and finite intersections, and be a compact paved space. Define to be the closure under finite unions and finite intersections of the paving on . For an -capacity , define
where is the projection map onto . We show that is a -capacity and that whenever .
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
|Title||proof of extending a capacity to a Cartesian product|
|Date of creation||2013-03-22 18:47:41|
|Last modified on||2013-03-22 18:47:41|
|Last modified by||gel (22282)|