dominated convergence theorem for convergence in measure

hi,

Ive just read this on wikipedia:

"(X, M, μ) - measure space. If μ is σ-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 --> 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 μ is σ-finite?

Thank you