2
$\begingroup$

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?

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

1 Answer 1

2
$\begingroup$

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

$\endgroup$
1
  • $\begingroup$ 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?). $\endgroup$ Sep 17, 2013 at 10:53

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Not the answer you're looking for? Browse other questions tagged or ask your own question.