>>10172338There is no contradiction. The scaling factor that appears in front of converges to 1 when p goes to infinity.
>>10172147Well I don't know about that, it looks suspicious (on [0,1], it would imply that all positive functions are a.e. equal to their sup). However, we have the next best thing.
Recall that, for each , we have , where is the Lebesgue measure.
I am not sure what your definition of is. Either this is by definition or, if you are working with continuous functions and defining it as the supremum, this should be an easy exercise.
Then, for each , set . We have:
Then, we have:
Letting go to infinity and recalling that , we get
This being the case for each , we get
The other inequality yields , which completes the proof.