In this topic, we prove the following statement
Statement: Let be a probability space. Let . Then is essentially bounded iff for all .
Proof. If is bounded, then by using the Holder inequality one has
for all . Conversely, we suppose is such that whenever . Let
Then partitions . Let
where it is understood that the -term is omiited if . Then
which implies . Since
where . Sincc we have that is finite and therefore is essentially bounded.