Hi,
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!
Thanks
Assume .
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.