Talk:Egorov's Theorem

The theorem, as presently stated here, is "on a set of finite measure, pointwise a.e. convergence implies uniform a.e. convergence".

This is not true. See for example $f_n (x) = n \chi_{[2^{-n}, 2^{-(n+1)}]}(x)$ on the set $[0, 1]$. This function converges pointwise, and thus pointwise a.e., but it does not converge uniformly a.e. (Credit to Terrence Tao for the counter-example.)

The proper statement of Egorov's Theorem is "on a set of finite measure, pointwise a.e. convergence implies almost uniform convergence." Lilred (talk) 08:15, 1 November 2016 (EDT)

Actually, if you follow the link to Definition:Uniform Convergence Almost Everywhere, you will see that it defines what you call "almost uniform convergence". Thus actually it is only the definition in need of rewriting, and not the theorem or its proof. — Lord_Farin (talk) 13:00, 1 November 2016 (EDT)

For what it's worth, "almost uniform convergence" is pretty standard terminology for this concept[1][2][3], while "uniform convergence almost everywhere" refers to something completely different[1][2]. So the proper course of action would be renaming the "Uniform Convergence Almost Everywhere" to "Almost Uniform Convergence", and fixing any articles that link there to use the updated terminology. That's a pretty big change though, and I'm new to ProofWiki, so I'm a bit chicken about doing it. Lilred (talk) 16:52, 1 November 2016 (EDT)

It is worth pointing out that Definition:Uniform Convergence Almost Everywhere has no source citations, and was written by someone who has not been active for many years. It would therefore make sense to do as you suggest -- but I recommend we find a hardcopy somewhere which gives the appropriate detail. (Nothing wrong with Wikipedia but that it is a tertiary source, and stackexchange is more-or-less endorsed by $\mathsf{Pr} \infty \mathsf{fWiki}$, but a published hardcopy is permanent.) Unfortunately I do not have enough background in Measure Theory to be able to take this task on. --prime mover (talk) 17:05, 1 November 2016 (EDT)

I've taken a cursory look around and I can't find a hard copy source. It's not in Shakarchi & Stein, not in Bartle & Sherbert, not in Rudin. There's a blog article by Terrence Tao, who is as good a primary source as any, but it's still not hard copy. Maybe an archived version of this post? Lilred (talk) 20:25, 1 November 2016 (EDT)

Job done. Dug out my copy of 1981: G. de Barra: Measure Theory and Integration which I have not got round to studying properly yet. It confirms Terry Tao.--prime mover (talk) 04:48, 2 November 2016 (EDT)