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). ◻