# Original source for a well-known result of convergence in measure and almost everywhere

A 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 $(f_n)_{n=1}^\infty$ convergences in measure to a function $f$ if and only every subsequence $(f_{n_m})_{m=1}^\infty$ has a subsequence $(f_{n_{m_p}})_{p=1}^\infty$ which converges to $f$ almost everywhere.

• $(1_{(n,\infty)})_{n=1}^\infty$ is the standard counterexample for the result you state. – Bill Johnson Sep 17 '13 at 13:53
• Since the other direction is a consequence of the metrizability of the space of real-valued measurable functions on $(X,\mathcal{A},\mu)$, one may look for whom introduced a distance on that space (Frechét?). – Pietro Majer Sep 17 '13 at 10:53