Time for some more measure theory. Given a measure space , let denote the set of measurable nonnegative real-valued functions.
- Let . Then .
- Let and . Then .
- Let with . Then .
- If and where we may assume WLOG
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 2 Let 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