Timeline for When can a sum and integral be interchanged?
Current License: CC BY-SA 4.0
17 events
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| Apr 17, 2023 at 2:33 | comment | added | Nate Eldredge | @TymaGaidash: Yes. Parametrize your contour and write the contour integral as the corresponding integral over an interval in $\mathbb{R}$. Then you can apply this result in its original form. | |
| Apr 14, 2023 at 11:20 | comment | added | Тyma Gaidash | @NateEldredge For the “99% of cases”, would Fubini’s theorem work for interchanging a contour integral with a sum? | |
| Dec 11, 2020 at 20:31 | history | edited | Michael Hardy | CC BY-SA 4.0 |
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| Sep 26, 2017 at 13:54 | comment | added | Nate Eldredge | The integral from $-\infty$ to $\infty$ is already handled above (i.e. it's when you work with Lebesgue measure on $\mathbb{R}$). The indefinite integral is a completely different situation and you should post it as a new question. | |
| Sep 26, 2017 at 13:53 | comment | added | Royi | @NateEldredge, Could you address the cases the integral is $ \int_{-\infty}^{\infty} $ and the case of indefinite integral? Thank You. | |
| Aug 15, 2017 at 20:58 | comment | added | Nate Eldredge | @User31443: see my edit. I guess there are a couple of standard facts that one needs to verify. | |
| Aug 15, 2017 at 20:58 | history | edited | Nate Eldredge | CC BY-SA 3.0 |
explanation
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| Aug 15, 2017 at 20:50 | comment | added | user389066 | @NateEldredge: I am trying to use the special case you mentioned for my query here: math.stackexchange.com/questions/2394761/… | |
| Aug 15, 2017 at 20:47 | comment | added | user389066 | @NateEldredge: I am unable to grasp how the general statement reduces to this special case if the measures are counting measure on $\mathbb{N}$ and Lebesgue measure in $\mathbb{R}$. Can you please elaborate that? | |
| Aug 15, 2017 at 20:44 | comment | added | Nate Eldredge | @User31443: I don't know of one. There is really nothing to show. What do you need a reference for? Is there some specific part you're not clear about? | |
| Aug 15, 2017 at 20:42 | comment | added | user389066 | @NateEldredge: I am looking for a reference for how the general statement implies the special case mentioned. | |
| Aug 15, 2017 at 20:38 | comment | added | Nate Eldredge | @User31443: Are you looking for a reference for the usual general statement of Fubini's theorem (Folland's Real Analysis, for instance) or for how the general statement implies this special case? It is literally just the general case applied with Lebesgue measure and counting measure. | |
| Aug 15, 2017 at 18:43 | comment | added | user389066 | Nate: Can you please point out a standard reference for "Fubini's theorem says that for general $f_n$, if $\int \sum |f_n| < \infty$ or $\sum \int |f_n| < \infty$ (by Tonelli the two conditions are equivalent), then $\int \sum f_n = \sum \int f_n$"? | |
| Nov 6, 2014 at 16:54 | comment | added | user3503589 | Ah that was my confusion. I was thinking of the case when $f_n(x) \geq 0$ and $\int \sum f_n(x) \to \infty$ and therefore Fubini theorem would be invalid(but luckily as you mentioned, its not"if and only if". Thank you for clearing it up Nate, though it was a stupid question. | |
| Nov 5, 2014 at 14:48 | comment | added | Nate Eldredge | Yes, that's what I said - I'm not sure what part is confusing you? Notice that I did not say "if and only if"! The two theorems give two different hypotheses, each of which leads to the same conclusion. (And note carefully the appearance of the absolute value bars in the hypothesis of Fubini's theorem.) | |
| Nov 5, 2014 at 9:24 | comment | added | user3503589 | I am a bit confused because first you say that $\sum \int f_n(x) dx=\int \sum f_n(x) dx$ holds if $f_n(x) \geq 0$which implies that even if the double integral is not finite the equality does hold and then you quote the Fubini theorem which says that the two integrals are equal if the double integral is finite. I would be grateful if you could explain this to me? Its probably as very stupid question but it makes me very uncomfortable | |
| Nov 19, 2011 at 21:32 | history | answered | Nate Eldredge | CC BY-SA 3.0 |