Convergence of increasing measurable functions in measure? Let ${f_{n}}$ a increasing sequence of measurable functions such that $f_{n} \rightarrow f$ in measure.
Show that $f_{n}\uparrow f$  almost everywhere
My attempt
The sequence ${f_{n}}$ converges to f in measure if for any $\epsilon >0$ there exists $N\in \mathbb{N}$ such that for all n>N,
$$
m (\{x: |f_n(x) - f(x)| > \varepsilon\}) \rightarrow 0\text{ as } n \rightarrow\infty.
$$
I think that taking a small enough epsilon are concludes the result since $f_n$ are increasing
Can you help me solve this exercise?
Thanks for your help

If $f_n$ converges to $f$ in measure, we have a subsequence $f_{n_k}$ converging to $f$ almost everywhere. As $f_n$ is increasing, we know that $f_n$ converges at every point or every subsequence increases to $\infty$. But as the subsequence $f_{n_k}$ converges to $f$ at almost every point, we have that $f_n$ converges to $f$ almost everywhere.
