Thomas E.
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 Jun 17 revised Sigma-field of a sequence of Random Variables added 120 characters in body Jun 17 answered Sigma-field of a sequence of Random Variables Jun 16 comment Form of $\sigma(X_n)$ and $\sigma(X_n)$-measurability @Justin: What I meant was, that $\sigma(\tilde{X})=\sigma(\{\{X_{n}\leq a\}:a\in\mathbb{R},\,\,n\in\mathbb{N}\})$, i.e. that $\{X_{n}\leq a\}$ form of sets are generators of $\sigma(\tilde{X})$. This is straight from the definition, as $\sigma(\tilde{X})$ is the smallest $\sigma$-algebra for which respect all the functions $X_{n}$ are measurable, and $X_{n}$ to be measurable is equivalent of saying that $\{X_{n}\leq a\}$ is a measurable set for all $a\in\mathbb{R}$ since $\{]-\infty,a]:a\in\mathbb{R}\}$ generates the Borel algebra of $\mathbb{R}$. Jun 15 comment About the Wasserstein “metric” By the way, here's some terminology explained: a $\it{transference\,\,plan}$ from $\mu$ to $\nu$ is a member of $\Pi(\mu,\nu)$. Intuitively, $d\pi(x,y)$ measures the amount of mass that $\pi$ transfers from $x$ to $y$ and $d(x,y)^{p}$ is the cost function. An optimal plan is such $\pi$ for which the infimum is reached in the definition of $W_{p}$, and for every other transference plan we have an inequality $\leq$. As mentioned above, given that $X$ is Polish guarantees the existence of optimal transference plans between any pair of Borel prob. measures. Jun 15 revised About the Wasserstein “metric” added one set for clarification. Jun 15 revised About the Wasserstein “metric” small typo Jun 15 revised About the Wasserstein “metric” added 11 characters in body Jun 15 comment About the Wasserstein “metric” A useful material as mentioned in my answer is the book of Cedric Villani "Topics in optimal transportation". Also one good source is Luigi Ambrosio's and Nicola Gigli's "User's guide to optimal transportation". It deals alot of nice results and detailed treatment of this topic. For instance, it considers the cases when $X$ is one of the following: Polish, geodesic metric space, Riemannian manifold, geodesic space with Alexandrov curvature (positive or nonpositive), and it defines a Metric Ricci curvature for a compact geodesic space $X$ through the study of its associated Wasserstein space. Jun 15 answered About the Wasserstein “metric” Jun 14 accepted Vitali set of outer-measure exactly $1$. Jun 13 comment Normal, Non-Metrizable Spaces @ChrisEagle: I guess it depends how you define normal. Some include Hausdorff in the definition. Jun 13 comment Vitali set of outer-measure exactly $1$. @AsafKaragila: Thanks for the help, I will do that. Jun 13 revised Basic fact in $L^p$ space added 31 characters in body Jun 13 answered Basic fact in $L^p$ space Jun 13 comment Basic fact in $L^p$ space What role does $q$ play? Jun 13 comment Basic fact in $L^p$ space Which one is the basic fact and what fact are you trying to prove? Jun 13 comment Vitali set of outer-measure exactly $1$. @AsafKaragila: Thanks. Do you have some references for this where I could possibly look it up? Jun 13 comment Vitali set of outer-measure exactly $1$. @GiuseppeVitali: Do you know if anyone has gone through the details of this construction and checked whether it works and gives the desired set? Jun 12 comment Collection of converging sequences determines the topology? No problem. There is still one $X$ left though :-) Jun 12 comment Collection of converging sequences determines the topology? @BrianM.Scott: You define $\tau_{2}$ by using the notion of $X$ and $D$ simultaneously. That is the reason why I'm asking.