extending a capacity to a Cartesian product
A capacity on a set can be extended to a set function on a Cartesian product simply by projecting any subset onto , and then applying the original capacity.
Theorem.
Suppose that is a paved space such that is closed under finite unions and finite intersections, and that is a compact paved space. Define to be the closure under finite unions and finite intersections of the paving on .
If is an -capacity and is the projection map, we can form the composition
Then for any , and is a -capacity.
This result justifies looking at capacities when considering projections from the Cartesian product onto . 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 “measure” or “outer measure”.
Recall that if is -capacitable then, for any , there is an such that and . However, and, by the above theorem, . This has the following consequence.
Lemma.
Let be -capacitable. Then, is -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 |