Suppose and are measurable spaces. Denote by the -algebra on generated by sets of the form where and . Let and be measures on and , respectively. It is easy to varify that defines a seminorm on a measurable functions with on for .
Proof. Let One can check that it contains all sets of the form where and . To prove the lemma, it suffices to show that is a -algebra, i.e., posses the following properties:
(a) is closed under intersection.
Let and . Take such that and such that . Then Therefore, .
(b) and implies .
This property follows from .
(c) If is an increasing sequence in , then .
Since converges to pointwise on , decreases to zero for all . Therefore, that is, as . ◻
Proof. Let be an increasing sequence of sets in with and . As decreases to zero pointwise on as , there is some such that . From the case of finite measure spaces, there is a vanishing outside such that . Therefore, .
(b) Approximate by simple functions and apply the result of (a). ◻