Possible Duplicate:
Generalisation of Dominated Convergence Theorem
Ive just read this on wikipedia:
"$(X, M, μ)$ - measure space. If $\mu$ is $\sigma$-finite, Lebesgue's dominated convergence theorem also holds if almost everywhere convergence is replaced by convergence in measure."
How could i go about proving this?
I know that if a sequence $f_n \to f$ in measure, then there is a subsequence which converges to $f$ a.e. I can apply the DCT on this subsequence, but how would i show the it works for the whole sequence? Also, how would i use the fact that $\mu$ is $\sigma$-finite?
Thank you