I have the following question and would certainly appreciate any help please : we are given a sigma-finite measure space X and we are given a sequence of measurable functions fn which converge almost everywhere to a function f, we are also given a measurable set of finite measure E in X.
and we are asked to prove the following:
without using dominated or bounded convergence theorem
3. Using what we have done in 1 and 2 we need to prove the dominated convergence theorem in X which is given as sigma finite
Any help would be great
Thanks to all helpers very much
- Asks us to prove that on E we have convergence in measure of the fn sequence.
- Where I get really stuck: I have the following addition to the question: we are given that for each \epsilon > 0 there exists a K>0 such that for all natural n's we have:
and we are asked to prove the following:
without using dominated or bounded convergence theorem
3. Using what we have done in 1 and 2 we need to prove the dominated convergence theorem in X which is given as sigma finite
Any help would be great
Thanks to all helpers very much