Proposition.For any measurable function and we have

Here means there are constants depending only on so that .

*Proof. * For each , define

Then for all . Also, for all . Define . Then and is a family of disjoint sets in . Then we have

Note that if and only if for all .

For the upper bound, we have

where .

For the lower bound, we have

So

Hence,

Take . Then we finally get .