I'm trying to show that if a sequence of decreasing, measurable function $f_n$ converges to $0$ in measure, then it converges pointwise to $0$ a.e.
I know how to prove that if $f_n$ converges to $f$ in measure, there exists a subsequence $\{f_{n_k}\}$ that converges to $f$ a.e. I know there is a condition that it is a decreasing sequence, and it converges to a constant $0$, but how does that change the fact that the whole sequence (not subsequence) of functions converges to the same value?