Time for some more measure theory. Given a measure space , let denote the set of measurable nonnegative real-valued functions.

Theorem 1

- Let . Then .
- Let and . Then .
- Let with . Then .

*Proof:*

- If and where we may assume WLOG
then

and similarly

Hence

so that

From this it follows that if and , then and hence by monotone convergence (to be proved later!)

- If then
Hence if , we have , and thus

- Since , this follows immediately from

One remark: For and we define

It follows from monotonicity that if with , then for , and hence

Now for the **monotone convergence theorem**:

Theorem 2Let be a measure space and a sequence of measurable real-valued functions satisfying , and pointwise. Then is measurable and

*Proof:* is a sequence of nondecreasing real numbers, and thus has a limit (possibly ). Since for each , we have and thus

For the reverse inequality, let and let be a simple function with . Set . Since and , we have . Hence for each ,

Letting we obtain

Since this is true for all , it remains true for , and as was arbitrary, taking the supremum over we have