Talk:Convergence in measure

Something is not quite right here: let X=R and f_n be the indicator function for the interval $[n,\infty )$; obviously the sequence f_n converges to zero (pointwise/everywhere), but it does not satisfy the condition given as the definition of convergence in measure. [Take $\varepsilon =1/2$; then all the sets have infinite measure and (therefore) do not tend to zero.]

This appears to contradict two assertions in the rest of the article: 1) the characterisation (in the sigma-finite case) of convergence in measure via a.e.-convergence; and 2) the characterisation of convergence in measure via pseudometrics. [In the example above, $\rho _{F}(f_{n},0)$ does tend to zero for every set of finite measure F.]

Perhaps one should say that a sequence f_n converges "locally in measure" to f if, for every measurable set of finite measure F

$\lim _{n\to \infty }\mu \{x\in F:\left|f_{n}(x)-f(x)\right|>\varepsilon \}=0$.

Then the issues raised above would seem to be resolved by replacing "convergence in measure" by "local convergence in measure".

Boy Waffle (talk) 20:09, 5 January 2008 (UTC)

Welcome, Mr. Waffle! It seems you're right. I remember proving at one time that a.e./L^p-convergence implies convergence in measure on [0,1], but I don't remember anything else well. I imagine it generalizes to finite measures The first section, before the topology, is cited from statements throughout Royden, but perhaps not correctly. I don't have the book with me, unfortunately. As for the pseudometrics, I never personally verified that the topology describes the convergence, but the concepts do have the same name. —vivacissamamente (talk) 02:27, 7 January 2008 (UTC)

Thanks for the welcome, Vivacissamamente! For the moment, unfortunately, I'm nowhere near a decent library, but I checked Fremlin online, and at one point he writes

warning! the phrase "topology of convergence in measure" is also used for [...some other topology...] I have seen the phrase "local convergence in measure" used for [...the topology described here...]

So, I've rather substantially re-written the article to account for the fact that different authors seem to use the same phrase to mean different things. I have also added that the dominated convergence theorem holds w.r.t. local convergence of measure (at least in the sigma-finite case); this is an essentially trivial corollary of its characterization in terms of a.e. convergence. Finally, I have temporarily deleted the line about "Cauchyness in measure" (which is nonsensical as it stands) and replaced it with a general comment about Cauchyness in any topology generated by pseudo-metrics. It seems likely to me that there are also global and local versions of this notion. [I will eventually do some homework on that score...]