This theorem can be found here: Th 22.E
Question 1): I think "fundamental a.e." in the second line of the proof should be "converges a.e.", because I can not derive from almost uniformly Cauchy to Cauchy a.e.. Am I right?
Question 2) Since $f_{n_k}(x)$ is not necessarily convergent for all x, the f defined in the 3rd line may not definable on the whole X and therefore can not be used in the definition of convergence in measure. I think we should extend its definition as follows: if $\lim\limits_{k \to \infty } f_{n_k}(x)$ exists, then f (x)=this limit as the text indicates, otherwise, f (x)=0. I can prove that this f (x) defined on the whole X is measurable, so we can apply Theorem 22.B to obtain that $f_{n_k}$ converges in measure to this f. Am I right? The definition of convergence in measure in Halmos' book is here: link text
Thanks!
