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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.

By whom and where was this theorem originally proved?

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  • $\begingroup$ $(1_{(n,\infty)})_{n=1}^\infty$ is the standard counterexample for the result you state. $\endgroup$
    – Bill Johnson
    Commented Sep 17, 2013 at 13:53

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That every sequence congerging in measure has an almust surely congerging subsequence was apparently first shown by Riesz in 1909 in "Sur les suites de fonctions mesurables". I don't know about the other direction, but Frechet is a likely culprit. Source

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  • $\begingroup$ 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?). $\endgroup$
    – Pietro Majer
    Commented Sep 17, 2013 at 10:53

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