Timeline for What is the role of of continuity in this proof of Kantorovich duality?
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| May 18, 2022 at 8:58 | comment | added | Steve | I think your proof works, it's just that at two points (verifying that $\Theta$ is lsc to apply duality, and veryfing $\Theta^*(\pi) = \int c \,d\pi$ for nonnegative $\pi$) is just easier for continuous cost, and thus its reasonable to make this assumption to focus on the big picture of the proof. | |
| May 15, 2022 at 13:28 | history | edited | Akira | CC BY-SA 4.0 |
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| May 15, 2022 at 11:33 | comment | added | user95282 | A good first step in understanding the K--R theorem is to prove it for finite metric spaces, so that one is not distracted by continuity, measurability etc. | |
| May 15, 2022 at 11:15 | comment | added | user95282 | Sorry, I meant my statement to be a comment, not an answer. To understand the Kantorovich--Rubinstein theorem, it's best to realize that its real essence has nothing to do with countable additivity of the measures, or continuity. At its core is a statement about (finitely additive) linear functionals. I don't have a reference handy, but it is a simple exercise, using e.g. the Hahn--Banach theorem, to prove it for finitely additive measures. From that then one gets versions with perfect, Radon, etc. measures, simply because if both marginals are such then so is the measure on the product. | |
| May 14, 2022 at 8:16 | history | edited | Akira | CC BY-SA 4.0 |
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| May 13, 2022 at 23:15 | history | asked | Akira | CC BY-SA 4.0 |