Consider the measure space and suppose is a sequence of non-negative measurable functions, such that for all .
First we prove that .
Since for all , we have and further .
Sending on LHS gives us the result.
Then we only need to prove that . In this regard, denote (which is measurable as the limit of measurable functions) and consider a simple function so that .
Now, for and some , define the increasing sequence of measurable sets . Since , we have for . Moreover, we have since .
Recall that the set function for is a measure on . We now see that , where we have used the fact that for the last equality. Taking the supremum over simple functions such that and the supremum over yields the required inequality.
- ↑ Gerald B. Folland, Real Analysis: Modern Techniques and Their Applications, second edition, §2.2