Talk:Triangle Inequality for Integrals

A short comment that $\int \text{Im} (\alpha f) = \text{Im} \left(\int \alpha f\right)$ vanishes by definition of $\alpha$ would be great. Correct nontheless.

Not quite sure what you mean -- can you elaborate? --prime mover (talk) 13:29, 26 April 2015 (UTC)

The proof is a bit weird, while not wrong. The integral is real so we're only talking about $\alpha = \pm 1$. (and $\map \Re {\alpha f} = \alpha f$) I have a feeling like the proof is for a more general context, maybe $f : X \to \C \cup \set \infty$ or something, but we don't have the setup for complex Lebesgue integrals yet. Proof 2 is a much more natural approach anyway. Caliburn (talk) 17:52, 27 September 2021 (UTC)

I think we should have two subpages, /Darboux Integral containing Absolute Value of Definite Integral and /Integral of Integrable Function containing the measure theoretic result. What do we think? I also have a broader point about Darboux vs Lebesgue integral that I'll make on the main page talk in a little bit. Caliburn (talk) 16:57, 21 October 2022 (UTC)

Yes but /Darboux Integral/Definite integral, isn't it? --Usagiop (talk) 17:13, 21 October 2022 (UTC)

The definiteness is part of the theorem, so it's pointless adding /Definite integral, isn't it? What am I missing, can this be applied to primitives? --prime mover (talk) 17:25, 21 October 2022 (UTC)

I thought /Darboux Integral includes the result for $\int_{\R^n}$ and /Darboux Integral/Definite integral includes the special case for $\int_\R$. Isn't it, Calibur? --Usagiop (talk) 17:57, 21 October 2022 (UTC)

Sorry, is the Darboux/Riemann integral on $\R^n$ not defined in $\mathsf{Pr} \infty \mathsf{fWiki}$ at all? Then we do not need the subsubpage. --Usagiop (talk) 18:03, 21 October 2022 (UTC)

If it's not there, add it. --prime mover (talk) 18:25, 21 October 2022 (UTC)

I thought it was but it looks like you're right. The results with integrals over $\R^d$ use Lebesgue integrals. Caliburn (talk) 21:06, 21 October 2022 (UTC)

I was thinking one dimensional when I said Darboux integral but I guess it's not much more work to do $d$-dimensional. It was on my to do list to prove the interplay between Darboux and Lebesgue integrals after which point I wasn't going to care much about the former. Though I still think results should be put up for it for those of early undergraduate level. Caliburn (talk) 21:04, 21 October 2022 (UTC)