Talk:Existence of Integral on Union of Adjacent Intervals

Regarding the presentation, I would prefer splitting the statement "$\int = \int + \int$, provided both exist" and the statement "$\int$ exists iff $\int + \int$ exists". I'm not sure whether these should be on the same page, but I'd say they do (perhaps using subpages). This type of proof abounds in measure theory and real analysis, so either we already thought of some good way to present this, or we should do this now so that we can mimic the approach for future applications. &mdash; Lord_Farin (talk) 09:46, 27 April 2015 (UTC)


 * I don't know if the following solves the problem, but I think it is a (small) improvement:




 * Then:
 * if $f$ is Riemann integrable on $\left[{a \,.\,.\, b}\right]$, $f$ is Riemann integrable on $\left[{a \,.\,.\, c}\right]$ and $\left[{c \,.\,.\, b}\right]$
 * and
 * If $f$ is Riemann integrable on $\left[{a \,.\,.\, c}\right]$ and $\left[{c \,.\,.\, b}\right]$, $f$ is Riemann integrable on $\left[{a \,.\,.\, b}\right]$
 * and in either case:
 * $\displaystyle \int_a^b f \left({x}\right) \ \mathrm d x = \int_a^c f \left({x}\right) \ \mathrm d x + \int_c^b f \left({x}\right) \ \mathrm d x$ --Ivar Sand (talk) 07:23, 28 April 2015 (UTC)


 * The usual way we state such a theorem as this is:


 * "$f$ is Riemann integrable on $\left[{a \,.\,.\, b}\right]$ $f$ is Riemann integrable on $\left[{a \,.\,.\, c}\right]$ and $\left[{c \,.\,.\, b}\right]$."


 * Then the proof is split into two parts: "Necessary Condition" and "Sufficient Condition". Whether those two separate parts are put in separate pages and then each transcluded depends on how complicated the proof of each part is. --prime mover (talk) 10:54, 28 April 2015 (UTC)


 * ... note we already have a page stating $\displaystyle \int_a^b f \left({x}\right) \ \mathrm d x = \int_a^c f \left({x}\right) \ \mathrm d x + \int_c^b f \left({x}\right) \ \mathrm d x$ so that does not need to be done here. In any case, it's a separate result and so belongs on a different page, by the nature of the structure of itself. --prime mover (talk) 10:56, 28 April 2015 (UTC)


 * I am aware of that page (I assume you are talking about Sum of Integrals on Adjacent Intervals), but that theorem does not fit because it requires the function to be continuous. --Ivar Sand (talk) 10:53, 29 April 2015 (UTC)


 * What I have gathered so far concerning what should (or could) be done:
 * - Write a new theorem "$f$ is Riemann integrable on [a..b] if and only if $f$ is Riemann integrable on [a..c] and [c..b].", and I propose the following name for it: Existence of Integral on Union of Adjacent Intervals. Its proof is based on the proof of the current theorem.
 * - The current theorem is changed to:
 * Let $f$ be a real function which is Riemann integrable on a closed interval $\left[{a \,.\,.\, b}\right]$, $a < b$.
 * Let $c$ be a point in $\left({a \,.\,.\, b}\right)$.
 * Then $\displaystyle \int_a^b f \left({x}\right) \ \mathrm d x = \int_a^c f \left({x}\right) \ \mathrm d x + \int_c^b f \left({x}\right) \ \mathrm d x$
 * - The proof of the current theorem is simplified by using references to the new theorem. --Ivar Sand (talk) 11:00, 29 April 2015 (UTC)


 * As I said, the page proving $\displaystyle \int_a^b f \left({x}\right) \ \mathrm d x = \int_a^c f \left({x}\right) \ \mathrm d x + \int_c^b f \left({x}\right) \ \mathrm d x$ already exists: Sum of Integrals on Adjacent Intervals. --prime mover (talk) 11:13, 29 April 2015 (UTC)


 * Your approaches can easily be unified by weakening the condition of Sum of Integrals on Adjacent Intervals, as the note there already suggests. &mdash; Lord_Farin (talk) 16:22, 29 April 2015 (UTC)


 * I think I could change the page Sum of Integrals on Adjacent Intervals by replacing the condition of continuity by a condition of integrability. However, I have a problem with its Sources section because here locally there is no library that contains the books I would need ... --Ivar Sand (talk) 11:32, 4 May 2015 (UTC)


 * My suggestion is to write a page with integrability instead of continuity. The existing page can stand, as a statement of the theorem for continuous functions, and it will have 2 proofs: Proof 1, which will say "as a continuous function is an integrable function, then ... (link to integrable function proof), and Proof 2, which is the existing poof. --prime mover (talk) 11:53, 4 May 2015 (UTC)


 * Separating the two seems natural, because continuity makes stuff a lot easier, allowing for more accessible proofs. On the other hand, we do want to document the most general result out there. &mdash; Lord_Farin (talk) 16:22, 4 May 2015 (UTC)


 * Exactly -- we want to document both, so that a) we have the most general result, and b) we have the accessible-to-freshmen version. --prime mover (talk) 19:08, 4 May 2015 (UTC)


 * I can see that the suggestion of writing a page with integrability seems natural in a general case. However, in this case, the proof of Sum of Integrals on Adjacent Intervals is basically a proof for a theorem for integrable functions already, and this is why the suggested weakening of the continuity condition to an integrability condition would, in my opinion, be best. It seems pointless to keep the continuity condition since it is easily removable.


 * However, it could be done as suggested, writing a page with integrability. Still, the fact that today's proof of Sum of Integrals on Adjacent Intervals is basically a proof for a theorem for integrable functions would manifest itself by Proof 2 and the proof for a theorem for integrable functions as being almost identical, and this would make it seem pointless to keep the existing proof as Proof 2.


 * To clarify: If I were to follow the suggestion of writing a page with integrability, I would like to do the following:


 * 1. First include in the proof of Sum of Integrals on Adjacent Intervals something like the following line, which I feel is missing today:
 * "As a continuous function is an integrable function, $f$ is integrable on any closed interval 𝕀." In addition, I would like to change some minor stuff in the proof at the same time. The Comment and Sources sections remain.


 * 2. Write a new page with integrability condition instead of continuity condition based on the page Sum of Integrals on Adjacent Intervals, like this:
 * - Remove the new line "As a continuous function is an integrable function, $f$ is integrable on any closed interval 𝕀." from the proof. The rest of the proof remains unchanged.
 * - Remove the continuity condition from the theorem Integral on Zero Interval, as that condition is unnecessary as far as I can see.
 * - The Comment section is removed.
 * - I am still uncertain what to do with the Sources section.
 * - Name for the new page: "Sum of Integrals on Adjacent Intervals for Integrable Functions". —Ivar Sand (talk) 10:11, 6 May 2015 (UTC)

Good suggestions. Some addenda: 1. The existing page should be renamend to reflect that it pertains only to continuous functions. Perhaps the two results should be transcluded onto a single "master page". 2. Simply place Template:SourceReview calls on the "Sources" sections. Prime.mover and I will sort them out. &mdash; Lord_Farin (talk) 19:23, 7 May 2015 (UTC)


 * What I feel remains to be done now for cleaning up the contents of the page Sum of Integrals on Adjacent Intervals in this context is:
 * 1. Remove the Expand template.
 * 2. Replace the lines in the proof "WLOG assume a<b." ... "Hence the result, from Proof by Cases." by: "The result follows by application of Sum of Integrals on Adjacent Intervals for Integrable Functions."
 * 3. Remove "Tom M. Apostol: Calculus Volume 1 (1967): §1.6" from the Sources section as that reference (§1.6 is called The concept of area as a set function) does not talk about integrability for continuous functions.
 * 4. Add a SourceReview template suggesting to remove the rest of the references in the Sources section if they do not talk about integrability for continuous functions. --Ivar Sand (talk) 08:34, 29 May 2015 (UTC)


 * Sounds good. Furthermore, 5. Rename the page with the addendum "for Continuous Functions".


 * 6. Remind me in due time to create the master transclusion page for both versions.
 * 7. Replace $\mathbb I$ with $I$ or something like that. The blackboard font $\mathbb I$ is not to be used for this purpose.


 * Thanks for your continued work on this :). &mdash; Lord_Farin (talk) 10:11, 30 May 2015 (UTC)