Timeline for Transitive closure of balanced mass transport in Z (move to close)
Current License: CC BY-SA 3.0
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| Apr 4, 2015 at 3:22 | comment | added | Christian Remling | @JamesPropp: This part is NOT vague: I'm describing a simple algorithm how to find the $\mu_j$, $\nu_j$ that you require to check that $\mu\sim\nu$. After the initial steps ($\nu_1=\delta_0+(1/2)\delta_3$, $\nu_2=(1/2)\delta_1$), I next obtain $\nu_3=(1/2)(\delta_1+\delta_5)$, $\nu_4=\delta_3$. The next two would be $\nu_5=(1/2)(\delta_3+\delta_7)$, $\nu_6=\delta_5$ etc., and of course you could state this as a general formula if you prefer, so I've defined all $\nu_j$. I've also simultaneously defined $\mu_j$'s with the required properties. | |
| Apr 4, 2015 at 3:21 | comment | added | James Propp | Thanks for providing more details! However, I am still not convinced. The part of the argument that I'd like to see elaborated is "So I can continue indefinitely in this style". Applying the move an indefinite but finite number of steps is fine; but it is not clear that performing the operation infinitely many times keeps one in the same $\approx$ equivalence class. To justify this one needs to show that the sequence of measures converges in the total variation metric, and that does not seem to be the case here. | |
| Apr 3, 2015 at 17:49 | history | edited | Christian Remling | CC BY-SA 3.0 |
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| Apr 3, 2015 at 17:26 | history | edited | Christian Remling | CC BY-SA 3.0 |
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| Apr 3, 2015 at 17:20 | history | edited | Christian Remling | CC BY-SA 3.0 |
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| Apr 3, 2015 at 2:00 | history | answered | Christian Remling | CC BY-SA 3.0 |