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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.
Jun
12
comment Collection of converging sequences determines the topology?
What role does $X$ play in the above construction. Is $D\subset X$?
Jun
12
revised Collection of converging sequences determines the topology?
added TeX.