Talk:Sum of Integrals on Adjacent Intervals for Continuous Functions

Your final "Comment" section reads like: "If we only had some eggs, we could have ham and eggs, if we only had some ham." --prime mover 18:20, 22 January 2012 (EST)
 * I do not like green eggs and ham. --GFauxPas 19:07, 22 January 2012 (EST)

I want to create a version of this theorem for complex integrals. My intention is to call the new theorem Sum of Integrals on Adjacent Intervals/Corollary, and write the theorem on a subpage of this page. Or, should I instead call the new theorem Sum of Complex Integrals on Adjacent Intervals? --Anghel (talk) 22:42, 5 December 2012 (UTC)


 * You could make the latter a redirect to the former, or, which I prefer, put it as a corollary on this page, but do not create it as a subpage, but properly on its own page with the suggested second title. Cf. Definition:Exponential (Category Theory) and Definition:Category with Exponentials. It is my experience that this practice makes web searches on PW more effective. --Lord_Farin (talk) 22:53, 5 December 2012 (UTC)


 * I disagree - making a subpage has the added advantage that a link to the parent page is available at the top of the page so you can directly go there and view the page in its full context - which you miss if you merely transclude a random page which is not a subpage. --prime mover (talk) 23:05, 5 December 2012 (UTC)


 * In this case the corollary is a proper generalization; I use this construction sparsely, when I feel an Also see would be too weak an indication of the connection of the results. In general, the construction you advocate is to be the default approach; I'm merely suggesting another approach may have its merit in certain cases. --Lord_Farin (talk) 23:34, 5 December 2012 (UTC)


 * No worries. Get the page written is the main thing, we can determine the best way to incorporate it into the existing structure in due course. --prime mover (talk) 06:44, 6 December 2012 (UTC)

Continuity
Why would we impose continuity in place of mere Riemann integrability? Same for Sum of Complex Integrals on Adjacent Intervals. --Lord_Farin (talk) 11:15, 8 December 2012 (UTC)


 * Because the initial page was written based on a child's first guide to analysis. That grown-up stuff about Riemann integrability wasn't covered. --prime mover (talk) 12:40, 8 December 2012 (UTC)


 * Answer to the second part of the question: Because Sum of Complex Integrals on Adjacent Intervals needed to reference this page in the proof. It isn't a great loss for complex analysists to assume continuitity of the integrable function. In fact, one of my main sources only defines the complex integral for continuous functions. --Anghel (talk) 22:05, 8 December 2012 (UTC)

More Generality
Does anyone know if this site has a version of the more general measure-theoretic version of this statement? I mean the fact that in a measure space $(X,\mu)$, with measurable sets $A \subset E$, and an integrable function $f$, the equality:
 * $\ds \int_E f \rd \mu = \int_A f \rd \mu + \int_{E \setminus A} f \rd \mu$

If so, what is the page called?

-- Ovenhouse (talk) 23:19, 30 December 2014 (UTC)


 * I'm pretty confident that it doesn't exist yet, because virtually all general measure theory is taken from Schilling's book, and he glosses over it.


 * If you can think of a good name, feel free to create the page; the proof is almost instantaneous using an appropriate "linearity for integrals" theorem. &mdash; Lord_Farin (talk) 14:20, 31 December 2014 (UTC)


 * Yeah, Rudin and Royden do the same thing. They skip the proof and say it's immediate. Although, yeah, it follows pretty quickly from using linearity. -- Ovenhouse (talk) 15:35, 31 December 2014 (UTC)