Definition talk:Definite Integral/Darboux

Renaming
I've reviewed the sources linked (except for Bell and Stewart) and half a dozen of my own, and it seems that, while many call this the "Riemann Integral", those that care to make the distinction, do call this the Darboux integral. I'd say, in the spirit of being unambiguous, it would be safe to change all instances of "Riemann" to "Darboux".

On the note of the maintenance tag, I had placed it myself and thought that goal had been accomplished. --Keith.U (talk) 08:07, 15 July 2016 (UTC)


 * Apologies -- hadn't realised you'd gone through the sources. It confused me because you'd left the SourceReview tag in there. And you say you hadn't done Bell and Stewart. The way this was planned (i.e. the way I have been using it) is to put everything that has been checked above the tag, and everything that hasnt been checked below. As you haven't checked BEll and Stwaret, best to leave them below the tag. I don't know. I'm sleeping badly at the moment and everything is becoming too great an effort to make sense of --prime mover (talk) 08:43, 15 July 2016 (UTC)


 * But the real problem I have is that many of the links back to this page still say "Definition:Definite Integral" which made me wonder whether this job has been done. --prime mover (talk) 08:44, 15 July 2016 (UTC)


 * Okay, sorry, one of them hadn't been done. --prime mover (talk) 08:47, 15 July 2016 (UTC)


 * Understood. This refactor has certainly stirred up more than I thought it would. --Keith.U (talk) 08:52, 15 July 2016 (UTC)


 * On the other note - is there any objection to changing instances of "Riemann Integral" to "Darboux Integral" on this page? A function on a closed bounded interval is Riemann Integrable iff it is Darboux integrable, so perhaps "Riemann-Darboux Integrable" is worth consideration. I have seen this, though not too often (it is a mouthful). The other option, from what I'm seeing, would be to change existing uses of "Riemann integrability" to "Darboux integrability" (in existing proofs, etc.), and in the future to choose Riemann or Darboux based on the definition used.


 * Either way, I'm gonna give the proverbial dust some time to settle first. --Keith.U (talk) 09:00, 15 July 2016 (UTC)


 * I have never seen the phrase "Darboux Integrable", it's always "Riemann Integrable". I suggest that we open a separate page: "Definition:Definite Integrable/Riemann Integrable" and include in it the simple statement that a function is "Riemann Integrable" iff blablabla, include it as a subpage of Definite Integral, and change the page "Definition:Riemann Integrable" to point to that page.


 * We still have to write the page proving the equivalence of Darboux Integral and Riemann Integral. Whether we also prove the fact that "Darboux Integrable" and "Riemann Integrable" are the same thing on that same page, or whether we raise it as a separate page (perhaps as a corollary) depends on the work involved.


 * There is also the task of going through all the links to "Definition:Definite Integral" and seeing whether there are any more instances of "Definition:Definite Integral|Riemann Integral", which was the product of someone who has been doing a lot of work in this area, but missed the point of how we have been using redirects. There may be other similar instances. That will be a job which may take some time, but is probably worth doing. --prime mover (talk) 10:14, 15 July 2016 (UTC)


 * And finally: as Riemann and Darboux are equivalent, I propose we just use "Definite Integral" when linking to this set of pages in the general run of things, and only refer to "Riemann Integral" when necessary to emphasise the fact that the internal structure of the thing is important. When just performing run-of-the-mill integration (take as an instance the Area inside Astroid, for example) then all we care about is that it is a "definite integral".


 * If we are in the field of measure theory, then we raise the concept of the Lebesgue integral, and differentiate it from the Riemann.


 * As for the Darboux Integral page (this one), we need to change the wording where it says "Riemann" and change it to say "Darboux" or it's confusing. I'll leave that up to you as you have the source works here. Most of my library does not focus on the fine details of real analysis. I'd have to do a lot of digging. --prime mover (talk) 10:20, 15 July 2016 (UTC)


 * How does User:Keith.U/Sandbox look? Obviously there is fleshing out to do on all the pages (I've taken a number of shortcuts before the structure is settled), but how is the exposition and structure? I saw your suggestion to open a page called "Definite Integrable," but this is not a phrase I was able to find in any of my sources. --Keith.U (talk) 05:02, 18 July 2016 (UTC)


 * Sorry, confusion. I meant "Definition:Definite Integral/Riemann Integrable" which makes sense as we already have the page "Definition:Definite Integral", and so the latter would be a subpage of that one. --prime mover (talk) 05:18, 18 July 2016 (UTC)


 * I tried tweaking the your sandbox page for Darboux Integral by removing the repetitive repetition of the repetitious bits. Since we have already got the definition of Riemann Integrable on the Definite Integral page, it seems silly to put it on there again.


 * We also need to build a page Equivalence of Definitions of Riemann Integrable which will be fun.


 * I also wonder whether we may want to hunt for a better way to present Definition 1, as the expression at the heart of it is long, unwieldy and not fully aesthetically pleasing, but I have to break off here and go to my day job. --prime mover (talk) 07:27, 18 July 2016 (UTC)


 * Shuffled the structuring a little to make a more concise reading experience while on the main page. Not sure how to obtain a similar result with nice transclusion unless we decide to go back to having the definitions of "integral" and "integrable" on the same page.


 * I have a proof of the equivalence, though I'll wait until the definitions are up and we can put some machinery in place.


 * Looked for other ways of presenting Definition 1. Found one source that writes it equivalently as the limit of the Riemann sums over a base for the partitions. Looking into this. --Keith.U (talk) 11:33, 18 July 2016 (UTC)


 * So every source I've found (checked some more at my university's library) presents it in the $\epsilon - \delta$ way we currently have. The one more concise notation I've seen (mentioned in passing) requires the more general definition of a limit over a filter, applied to $\R$. --Keith.U (talk) 13:31, 18 July 2016 (UTC)