So yeah my school makes us take a mini real analysis course for physics and I am really stumped on this one: 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.
I am asked to prove that on E we have convergence in measure of the fn sequence. I managed this.
Where I really get 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: $$\int |(f_n(x))| \chi_{{f_n>=K}} d \mu < \epsilon $$
and we are asked to prove the following: $$\lim_{n\to\infty} \int |(f_n(x)) - f(x)| d\mu$$
without using dominated or bounded convergence theorem
- 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