As far as I know, it takes a fair bit of machinery to prove such results, including Fubini's and Hölder's theorems and the duality between and . To simplify things, I'll just indicate how to do it when all the functions are defined on the whole of . You can then modify the proof to deal with domains that are subsets of .

So suppose that , and . Denote by the function . Then and .

Let . Then

Since that holds for all , it follows from duality that , with