It's well known that the doubling condition may not hold on $CD(K,\infty)$ space. Can one give an example such that: $(X,d)$ is a compact metric space, $\mu$ is a Borel probability measure and $(X,d,\mu)$ is a $CD(K,\infty)$ space, but the doubling condition fails?
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2$\begingroup$ Hilbert cube. (too short for an answer) $\endgroup$– Anton PetruninCommented Jan 20, 2016 at 13:29
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$\begingroup$ @AntonPetrunin:What's the probability measure $\mu$? Why does this space satisfy $CD(K,\infty)$? $\endgroup$– oneyearCommented Jan 20, 2016 at 13:42
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2$\begingroup$ Take the product measure, $CD(0,\infty)$ can be checked directly, or you may think of it as a limit space of n-dimensional rectangles for $n\to\infty$. $\endgroup$– Anton PetruninCommented Jan 20, 2016 at 13:54
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