# Original source on a result of convergence in measure and a.e.

A relatively well-known result in measure theory states that given a sequence $(f_n)_{n=1}^\infty$ of measurable functions from a $\sigma$-finite measure space $(X,\mathcal{A},\mu)$ to $\mathbb{R}$ then the sequence $(f_n)$ convergences in measure to a function $f$ if and only every subsequence $(f_{n_m})$ has a subsequence $(f_{n_{m_p}})$ which converges to $f$ almost everywhere.

In fact, a part of the proof of this fact is presented in this post.