1
$\begingroup$

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

  • 0
    On finite measure spaces almost everywhere convergence implies convergence in measure by Egoroff's theorem. Split $X$ into countably many finite measure spaces.2012-10-31
  • 0
    thank you http://math.stackexchange.com/users/9849/davide-giraudo2012-10-31

0 Answers 0