There's a decent chance you might find what you're looking for here. Maybe your library has it.
If is a sequence in converging in to an f in , that is, as , do we necessarily have ? If not, could you please come up with a counterexample?
If is a sequence in satisfying where , do we necessarily have converges in to f (as defined above)? If not, could you please come up with a counterexample?
Thanks!
No, I don't have this book. I searched just now the contents of this book from google books and the item 40 "sequences of functions converging in different senses" might be relevant, but I didn't find any useful information regarding my question from the pages that are allowed to read by google books.
Look at the reverse triangle inequality,
Take the real line with the Lebesgue measure and let . They have but it does not converge to that.If is a sequence in satisfying where , do we necessarily have converges in to f (as defined above)? If not, could you please come up with a counterexample?
Thanks!
Brilliant! Thank you very much, Focus!
So necessarily implies , but the converse does not. Sometimes the converse may be true, sometimes may not as your counterexample indicates. Only when it is known in advance that converges a.e. can the converse (convergence in ) necessarily hold.