Reading through a proof in a book that the intersection of Lp and Lq spaces is complete, I'm wondering the following (which I think was used by the author):
In a measure space , with , is it true that if is a sequence in such that in and in , then -a.e.
Since there is no relationship between the two norms for arbitrary spaces X, I do not know how to go about proving it if it were indeed correct. Your insights would be most appreciated!
You can prove this by noting that both convergence in and imply convergence in measure: for every , we have
(in probability, this is called Markov inequality). Thus,
and similarly . Since , this implies that . Since this holds for every , we have , i.e. a.e..
If , the convergence in measure is still true, in a stronger sense: for every , for large . In particular, we can conclude like above.