preservation of uniform integrability
Let be a measure space. Then, a uniformly integrable set of measurable functions will remain uniformly integrable if it is enlarged by various operations, such as taking convex combinations and conditional expectations. The following theorem lists some of the operations which preserve uniform integrability.
Suppose that is a bounded and uniformly integrable subset of . Let be the smallest set containing such that all of the following conditions are satisfied. Then, is also a bounded and uniformly integrable subset of .
is absolutely convex. That is, if and are such that then .
If and then .
is closed under convergence in measure. That is, if converge in measure to , then .
If and is a sub--algebra of such that is -finite, then the conditional expectation is in .
To prove this we use the condition that the set is uniformly integrable if and only if there is a convex and symmetric function such that as and
is bounded over all (see equivalent conditions for uniform integrability). Suppose that it is bounded by . Also, by replacing by if necessary, we may suppose that . Then, let be
which is a bounded and uniformly integrable subset of containing . To prove the result, it just needs to be shown that is closed under each of the operations listed above, as that will imply .
First, the convexity and symmetry of gives
for any and with . So, . Similarly, if and then and, .
Now suppose that converge in measure to . Then Fatou’s lemma gives,
Finally suppose that and . Using Jensen’s inequality,
|Title||preservation of uniform integrability|
|Date of creation||2013-03-22 18:40:20|
|Last modified on||2013-03-22 18:40:20|
|Last modified by||gel (22282)|