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 13h comment Representation of a linear functional Lipschitz in total variation Thanks, this example works perfectly. No weak continuity assumptions, I'm also not sure why did I go for Borel space at the time I've asked this question, as I do not seem to use this structure at all. Do you think weak continuity would make this true, or just harder to find a counterexample (in case you have already thought of that)? 1d accepted Representation of a linear functional Lipschitz in total variation May 3 comment Representation of a linear functional Lipschitz in total variation The answer below does not seem to be correct according to its author. May 1 awarded Notable Question Apr 28 revised Representation of a linear functional Lipschitz in total variation added 191 characters in body Apr 25 awarded Popular Question Apr 24 asked Minimizing the average Apr 24 revised Can anyone clarify the meaning of zero content? added 61 characters in body Apr 23 comment What is the optimal path between $2$ fixed points around an invisible obstructing wall? Can you tell where do you know this puzzle from? Apr 11 comment Decaying way to compute sample mean I think an easier check is to see that variance of $s_n$ has a positive limit in the end, whereas if $a_n = \frac1n$ as in usual sample mean, then variance converges to zero. Apr 11 revised Decaying way to compute sample mean added 213 characters in body Apr 9 comment Compact but not measurable Three upvotes? ${}$ Apr 9 asked Decaying way to compute sample mean Apr 9 awarded Favorite Question Apr 8 comment Extension of Kakutani's fixed point theorem. One obvious condition I can think of is: any point from the boundary is mapped to the inside. Apr 8 comment Approximate partition of unity by characteristic functions By characteristic functions you mean indicators, that equal $1$ on the set and $0$ otherwise? If yes, which kind of convergence are you talking about? Let's say, on real line $v_1 = \frac12(1 + \sin x)$ and $v_2 = 1 - v_1$, how would you construct a sequence of sets $E_{1j}$ that converge to $v_1$? Apr 8 awarded integration Apr 5 awarded Socratic Apr 4 comment Coarsest filtration $\mathcal F$ here is the $\sigma$-algebra on the original probability space, so yes, $\xi_t$ is $\mathcal F$-measurable for all $t$. W.r.t. existence, the shift process on $\omega$ satisfies all required properties. However, it is a very expensive process in terms of filtration. Apr 4 comment Coarsest filtration @TheBridge: I still don't get it, do you mean that for such process there does not exist the coarsest Markov super-pricess?