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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

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LFRC
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  • On finite measure spaces almost everywhere convergence implies convergence in measure by Egoroff's theorem. Split $X$ into countably many finite measure spaces. – Beni Bogosel Oct 31 '12 at 20:13
  • thank you http://math.stackexchange.com/users/9849/davide-giraudo – LFRC Oct 31 '12 at 20:17

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