Talk:Uniform integrability

The "measure-theoretic" definition
This appears unfortunate to me, not universally adopted (consider e.g. the two definitions, equivalent among themselves but different from the present one, in the books by Bauer and by Elstrodt, the latter apparently available in German only), and to weak to actually yield the Dunford-Pettis theorem given at the end: Under the definition as stated, for example the set of all indicators of bounded intervals in the real line would be uniformly integrable with respect to Lebesgue measure, but clearly not compact in the weak topology of $$\mathrm{L}^1$$.

I think this article should be rewritten, starting with the good definitions (making e.g. the Dunford-Pettis theorem true) from the books just indicated by me, then admitting that some books use weaker definitions, and explaining the problems with those. Lutz Mattner (talk) 17:19, 22 June 2019 (UTC)

UI martingales
I don't see anything on Wikipedia on the special importance of UI martingales. Not here, not in Martingale (probability theory), not in Optional stopping theorem. This is really a shame, and it is perhaps even more of a shame that I am willing to moan about it here but not willing to do anything about it. The book of Williams and the books of Rogers and Williams would be potentially good references if someone were to care more than me.

PS, that example of a non-UI sequence that needs a citation is probably in Williams but I don't have the book with me... 189.187.87.160 (talk) 00:52, 9 September 2012 (UTC)

Relation to a type of "pre-compactness" of a set of functions on the sigma-algebra
If $$\{X_c:c\in C\}$$ is a uniformly integrable set of random variables, consider for each $$c\in C$$, the function $$f_c:\mathcal{F}\rightarrow \mathbb{R}$$ defined by $$f_c(A):=E[|X_c|1_A]$$. Then the set $$\{f_c:c\in C\}$$ is something like "pre-compact" assuming a certain (metric?) topology on $$\mathcal{F}$$, since for any $$A,B\in\mathcal{F}$$, $$\underset{c\in C}{\sup}|f_c(A)-f_c(B)|\leq\underset{c\in C}{\sup}E[|X_c|1_{A\Delta B}]$$, which is small for small $$P(A\Delta B)=P(A\cup B\smallsetminus A\cap B)$$ (i.e., we have "equicontinuity", or bounded oscillations), and $$\underset{c\in C, A\in \mathcal{F}}{\sup}|f_c(A)|\leq \underset{c\in C}{\sup}E[|X_c|]<\infty$$ (pointwise boundedness). This can be a useful way of thinking about uniform continuity that I haven't really read much about, although Lemma II.20.7 in Volume 1 of Rogers and Williams' book (Diffusions, Markov Processes and Martingales) is presented in a way that induces the reader (at least me) to realize this.Vinzklorthos (talk) 03:28, 26 June 2012 (UTC)

Measure theory vs. probablity theory language
This article should be rewritten in the general language of measure theory (e.g., talk about functions instead random variables). asmeurer ( talk  |  contribs ) 05:06, 3 December 2012 (UTC)

Merger proposal : "uniform absolute continuity" into "uniform integrability"
I propose that uniform absolute continuity be merged into uniform integrability. The former is just a non-standard name for the latter. (The definitions on the two pages are identical.) Confusion does arise, since absolute continuity and uniform integrability are related but distinct: see the 2015 paper by Fitzpatrick & Hunt in the American Mathematical Monthly 122(4): http://dx.doi.org/10.4169/amer.math.monthly.122.04.362 — Preceding unsigned comment added by Mechanismus (talk • contribs) 21:44, 14 February 2018 (UTC)