Talk:Set-theoretic limit

Definitions
The first definition guarantees a limit, that limit being "at least" the empty set, {}. The second definition seems to suggest that it is possible for the limit superior to be different from the limit inferior, in which case the limit does not exist? Is it possible for the lim. inf. to be unequal to the lim. sup.? Doesn't this contradict the fact that the first definition guarantees the existence of a limit? Or is it that it is impossible for the two to be unequal? If so, that should be mentioned to avoid confusion, and perhaps the definitions should be presented as three different definitions: one using indicator functions, one using lim inf, and the other using lim sup. Or maybe the two definitions (one using indicator functions, and the other using lim inf and lim sup) are entirely different things, in which case this different should be highlighted, again, to avoid confusion.

EDIT: In fact, it has been shown to me that the first definition (using indicator functions) is equivalent to the given definition for "lim inf", and that indeed "lim sup" and "lim inf" can be different. So perhaps you should simply give one definition for the limit, that being the lim inf and the lim sup ONLY WHEN they are equal, and offer the characteristic function definition as an equivalent way to compute "lim inf." I would change it myself, but I'd rather have someone more knowledgeable actually do it (I don't actually know what the definition for a set-theoretic limit is, I just noticed the inconsistencies in the given ones).


 * This has been dealt with now. I have also reorganized the material to make this more like the typical Wikipedia article. Daren Cline (talk) 16:09, 9 April 2015 (UTC)

Please note that there is a more general sense for convergence of sets that depends on the topology or metric on X. I'm not particularly familiar with it, but have added a link to it. (Feel free to clarify my comment further.) Although the definition here is mentioned on that page (via the discrete metric), it does not cover it thoroughly, including its connection with indicator functions and its uses (e.g., in measure theory and probability), and the examples provided are pretty elementary and not especially illuminating. Moreover, that article is exceedingly long as it is. So I feel that it is quite important to have this separate article. Daren Cline (talk) 16:09, 9 April 2015 (UTC)

Expanding the Article
I'm intending to add a few examples (very) soon. I'd like to put in several more that aren't as simple; maybe with multidimensional sets.

I will also add to the properties section, including sketches of proofs.

I hope to provide uses in probability. Perhaps in measure theory too, but others more familiar may wish to do so - or even to replace mine. Daren Cline (talk) 16:09, 9 April 2015 (UTC)


 * These have been done, but I'm sure other (and perhaps more interesting) examples could be provided. Daren Cline (talk) 21:41, 10 April 2015 (UTC)

Event of limsup vs limsup of events
In the article under "almost sure convergence" it says: "It would be a mistake, however, to write this simply as a limsup of events". I think it would be very illuminating to discuss the difference between the event {liminf} and liminf{}, and how that could lead one astray, perhaps by expanding on this sentence. — Preceding unsigned comment added by Diego898 (talk • contribs) 18:17, 2 February 2016 (UTC)


 * I added a sentence explaining what the event is not. Daren Cline (talk) 18:30, 2 February 2016 (UTC)

Event of limsup vs limsup of events
This comment is related to the one of Diego899 Y(n) converges (Almost surely) to Y means that for each omega, for each epsilon, there is a N such that if n > N, |Y(n)[omega] - Y[omega]| < epsilon Does not it correspond exactly to the set intersection_{p > 1} union_{n > 1} intersection_{m > n} { |Y(m) - Y| < 1/n } I don't understand why it's not true — Preceding unsigned comment added by 194.153.106.250 (talk) 16:08, 21 December 2021 (UTC)