real analysis - Monotone convergence theorem assuming convergence in measure

I have heard that the monotone convergence theorem holds if the hypothesis of almost everywhere convergence is replaced by convergence in measure.
I concur; if $f_n$ converges in measure then there exists a subsequence of $f_n$ which converges to $f$ a.e. so that we can apply MCT for that subsequence but I couldn't see why the conclusion holds for the original sequence $f_n$.