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Questions tagged [polish-spaces]

For questions involving Polish spaces, that is, separable and completely metrizable topological spaces.

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When is this closed set compact

Apparently in the polish space $^\omega\omega$ a closed $K\subset\hspace{1mm}^\omega\omega$ is bounded and therefore compact if it is completely below some $f\in \hspace{1mm}^\omega\omega$ as in $K= \{...
L. R.'s user avatar
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Clarification on Lemma regarding Polish spaces [duplicate]

I read that if $X$ is a Polish space (i.e. a complete separable metrizable space), if $U \subseteq X$ is open and $\epsilon > 0$, then there are open sets $U_0,U_1,U_2, \dots$ s.t. $U = \bigcup U_n ...
Link L's user avatar
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Suslin measurable sets and the smallest field containing all analytic sets

Let $X$ be a Polish space. Recall that the Suslin operation is the operation $\mathcal{A}$ such that for any Suslin scheme $\{A_s : s \in \omega^{<\omega}\}$ of subsets of $X$, we have: $$ \mathcal{...
Clement Yung's user avatar
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3 votes
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Extension of Borel map from a separable metric space to a Polish space

Suppose that $f:X\to Y$ is a Borel map from separable metric space $X$ to a $T_3$ space $Y$. Does there always exist a Polish space $\tilde X \supseteq X$ and $T_3$ space $\tilde Y\supseteq Y$ and an ...
Jakobian's user avatar
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Axiom of Choice and Borel determinacy for Polish space

Given a set $A$, Borel determinacy for $A$ is the theorem (of $\mathsf{ZFC}$) asserting that every Borel subset of $A^\omega$ is determined. That is, if I and II take turns playing members of $A$, and ...
Clement Yung's user avatar
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Locally compact Polish space admits a proper metric [duplicate]

If $X$ is locally compact Hausdorff, then the following are all equivalent: $X$ is second countable, $X$ is metrizable and $\sigma$-compact, $X$ is metrizable and separable, $X$ is Polish. I want to ...
subrosar's user avatar
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3 votes
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Why is the set of probability measures not weak*-compact?

Let $M(X)$ be the set of probability measures on a Polish space $X$ with Borel $\sigma$-field. Further consider the properties of $M(X)$ when considered as members of the dual space of $Y:=C_b(X)$ - ...
P.Jo's user avatar
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Functional E convex and lower semicontinuous implies weakly lower semicontinuous in Wasserstein space

I have a certain functional $E : W_2 \rightarrow \mathbb{R}$, where $W_2$ is the 2-Wasserstein space (metric and separable). Such functional is convex. Now, can I state that if $E$ is strongly lower ...
Erwin Smith's user avatar
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Sufficient Conditions on Metric Space for Wasserstein Distance?

For a metric space $M$ and the Wasserstein 1-distance $W_1$, what qualifying assumptions do we need for $M$? I have seen we only need $M$ be compact, $M$ be a Polish space, or $M$ be a Radon space. ...
stone327's user avatar
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Comparing sigma-algebras generated by a random variable and its induced posterior

Consider two random variables $\tilde x$ and $\tilde y$ taking values in Polish spaces $X$ and $Y$ respectively. Let the (prior) distribution of $\tilde x$ be $\nu$ and the distribution of $\tilde y$ ...
Y Ava's user avatar
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Total variation distance: a relationship between a Polish space $(\mathcal{X}, d)$ and a measurable space $\left(\mathcal{X},\mathcal{A}\right)$

Introduction (part 1). In the following excerpts of Villani (2008) Optimal transport, old and new, Villani (i) defines the Wasserstein distance among two probability measures $\mu$ and $\nu$ on a ${\...
Ommo's user avatar
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Refine product topology to make Borel sets be clopen

I'm working on Exercise 2.28 in Prof. David Marker's notes http://homepages.math.uic.edu/~marker/math512/dst.pdf on refining the topology to make Borel sets clopen. Question: Suppose $X$ is a Polish ...
Hans's user avatar
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1 answer
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Can every open set of a polish space be expressed as a countable union of compact sets?

Let $X$ be a polish space and $A \subset X$ be an open subset. Can we find a sequence of compact subsets $(K_n)_{n \in \mathbb{N}} \subset X$ such that $A= \cup_{n \in \mathbb{N}} K_n$? I read a post ...
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Subspace of Polish space is Polish

Let $X$ be a Polish space. We know that $A\subset X$ is Polish iff it is a $G_\delta$ subset. So, since every open and every closed subset is $G_\delta$, it is in particular Polish. Now, we let $A\...
math_as_a_lifestyle's user avatar
3 votes
1 answer
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Clarification about weak topology in the space of probability measure

In Jacod and Shiryaev book, page 347, we find the definition of weak convergence of probability measures. Definition. Let $E$ be a Polish space (completely metrizable space which is also separable) ...
AlmostSureUser's user avatar
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Generalization of the L^p space?

I've recently had a look at https://helios2.mi.parisdescartes.fr/~jdedecke/p1.pdf . In chapter 3, Definition 3.1, they defined: For any $p \geq 1$, let $\mathbb{L}^p$ be the class of real-valued ...
mathrunner's user avatar
2 votes
1 answer
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Locus of Continuity of a Function between Polish Spaces

I think the question is maybe a bit trivial, but right now my head doesn't work that much well so I'm asking y'all to have some clarification. I know that, for a subset $A\subseteq \mathbb R$, the ...
alvoi's user avatar
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Definition of Polish space: why homeomorphic?

While glancing over measure theory books I noticed a discrepancy in the definition of a Polish space: given a topological space $(X,\mathcal T)$, some authors use Definition A: $X$ is a Polish space ...
Marlou marlou's user avatar
3 votes
1 answer
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Compact vs. Polish space in application field.

I was asked in a presentationto give an example of an engineering system where you want to use a system's state space as a Polish space but not a compact space. The audience was all electrical ...
Myshkin's user avatar
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1 answer
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The existence of a $\sigma$-compact set with full probability

Let $X,Y$ be Polish spaces and $\mathcal P(X)$ the space of all Borel probability measures on $X$. Fix $\mu\in \mathcal P(X), \nu \in \mathcal P(Y)$. Let $\gamma \in \Pi(\mu, \nu)$, i.e., $\gamma \in \...
Akira's user avatar
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2 votes
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How to construct a measurable map $\psi'$ such that $\psi' = \psi$ $\nu$-a.e.?

Let $X,Y$ be Polish spaces and $\mathcal P(X)$ the space of all Borel probability measures on $X$. Fix $\mu\in \mathcal P(X)$ and $\nu \in \mathcal P(Y)$. Let $\gamma \in \Pi(\mu, \nu)$, i.e., $\gamma ...
Akira's user avatar
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How does the use of disintegration theorem make sense here?

Let $X,Y$ be Polish spaces and $\mathcal P(X)$ the space of all Borel probability measures on $X$. Fix $\mu\in \mathcal P(X), \nu \in \mathcal P(Y)$. Let $\gamma \in \Pi(\mu, \nu)$, i.e., $\gamma \in ...
Akira's user avatar
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3 votes
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How to prove these definitions of an analytic set are equivalent?

Let $F$ be a set and $\mathcal F$ a collection of subsets of $F$ such that $\emptyset \in \mathcal F$. We denote by $F_\sigma$ (resp. $F_\delta$) the closure of $F$ under countable union (resp. ...
Analyst's user avatar
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1 answer
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Disintegration theorem: how do the authors prove that $\mu_y$ is supported on $\pi^{-1} (y)$ for $\nu$-a.e. $y \in Y$?

Recently, I came across Tao's blog about disintegration theorem. Disintegration theorem Let $X$ be a Polish space, $\mathcal X$ its Borel $\sigma$-algebra, and $\mu$ a Borel probability measure on $...
Analyst's user avatar
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Where completeness and separability are used in a proof of this lemma?

I'm reading Lemma 2.30. from this note. Lemma 2.30. Let $\mathcal{V}$ and $\mathcal{D}$ be two algebras of subsets of a separable complete metric space $T$. Assume $\mathcal{V} \subseteq \mathcal{D}$ ...
Analyst's user avatar
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Is Homeo$(X)$ a $G_\delta$ subset of $C(X,X)$?

I need to prove that if $X$ is a compact Polish space, then $Homeo(X)$ (the set of homeomorphisms from $X$ to $X$) is a $G_\delta$ subset of $C(X,X)$ (the space of continuous functions with de ...
david medina's user avatar
1 vote
1 answer
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Examples of dense and codense $G_\delta$ subsets of $\mathbb{R}^2$ that are not homeomorphic to $\mathcal{N}=\mathbb{N}^\mathbb{N}$

I have been asked to show that if $X$ is a dense $G_\delta$ subset of $\mathbb{R}$ such that $\mathbb{R}\setminus X$ is also dense in $\mathbb{R}$, then $X$ is homeomorphic to $\mathcal{N}=\mathbb{N}^\...
closedrhombus's user avatar
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1 answer
71 views

Let $\mu$ be a Borel measure and $x_n \to x$. Is it true that $\mu (B(x_n, r)) \to \mu (B(x, r))$?

Let $(E, d)$ be a Polish space and $\mu$ a finite Borel measure on $E$. Let $r>0$ and $x, x_n \in E$ such that $x_n \to x$. Is it true that$$ \mu (B(x_n, r)) \to \mu (B(x, r))? $$ My attempt: Let ...
Akira's user avatar
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0 votes
1 answer
98 views

On morphisms and equivalence of Polish Spaces

I've been wanting to start learning about Polish spaces because of their relationship with classification theorems. Polish Space: A space homeomorphic to a seperable complete metric space; source ...
ZFCarla's user avatar
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1 vote
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65 views

Is the $c$-transform $f^c$ defined by $f^c(y) := \inf_{x\in X} [c(x, y)- f(x)]$ measurable?

Let $X, Y$ be Polish spaces and $c:X \times Y \to [0, +\infty)$ lower semi-continuous. Assume that $f:X \to \mathbb R \cup\{\pm\infty\}$ is measurable. We define the $c$-transform $f^c:Y \to \mathbb R ...
Akira's user avatar
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1 vote
0 answers
23 views

Homeomorphism between $\mathbb N^\infty$ and a closed subset $\mathbb M$ of $(\mathbb N^\infty)^\infty$

Trying to figure out a proof of a lemma that I'm reading in Stochastic Relations by Ernst-Erich Doberkat. The Baire space, denoted $\mathbb{N}^\infty$, is the infinite product of the natural numbers. ...
Erik Grnl's user avatar
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1 answer
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If $c_n \nearrow c$ then $\lim_n \inf_{\pi \in \Pi(\mu, \nu)} \int c_n d\pi = \inf_{\pi \in \Pi(\mu, \nu)} \int c d\pi$

Disclaimer This thread is meant to record. See: SE blog: Answer own Question and MSE meta: Answer own Question. Anyway, it is written as problem. Have fun! :) Let $X,Y$ be Polish spaces and $c:X \...
Akira's user avatar
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2 votes
1 answer
282 views

Locally compact Polish space is $\sigma$-compact?

I have recently encountered this result. Let $X$ be $\sigma$-compact, locally compact Hausdorff space and $\mu$ is a Radon measure on $X$. Then the space of continuous functions with compact support ...
Akira's user avatar
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1 vote
1 answer
255 views

Let $X$ be a locally compact Polish space. Is the space of continuous functions with compact support dense in that of $\mu$-integrable functions?

I'm reading this question for which I would like to clarify the theorem mentioned there. We have (S1) Let $X$ be a locally compact Hausdorff space. Then the space of continuous functions with compact ...
Akira's user avatar
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1 vote
1 answer
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Let $X,Y$ be Polish spaces and $\mu, \nu$ Borel probability measures on $X,Y$. Then the coupling $\Pi(\mu, \nu)$ is uniformly tight and weak* closed

I'm trying to prove this result in Optimal Transport. Could you verify if my attempt is fine? Let $X,Y$ be Polish spaces and $\mu, \nu$ Borel probability measures on $X,Y$ respectively. Let $\Pi(\mu, ...
Analyst's user avatar
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2 votes
1 answer
494 views

Every finite Borel measure on a separable metric space is tight.

I'm trying to prove this property of Polish space. Could you verify if my attempt is fine? Let $(X, d)$ be a complete separable metric space. Then every finite Borel measure on $X$ is tight. I post ...
Analyst's user avatar
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4 votes
1 answer
82 views

How to prove that the following metric induces the subspace topology?

I am trying to follow Theorem (3.11) of Kechris's Classical Descriptive Set Theory. In this part of the proof he shows that a $G_{\delta}$-subspace Y of a completely metrizable space $(X,d)$ is ...
a_hayler's user avatar
1 vote
1 answer
109 views

Hausdorff Quasi-Polish spaces

A topological space is said to be quasi-Polish if it is second-countable and completely quasi-metrizable (see for an introduction de Brecht's article de Brecht, Matthew, Quasi-Polish spaces, Ann. Pure ...
Lorenzo's user avatar
  • 2,601
1 vote
0 answers
88 views

A continuous function with uncountable image between Polish spaces is injective on a generic compact set

Exercise 8.8ii in Kechris Classical Descriptive Set Theory asks to prove that if $f\colon X\to Y$ is a continuous function between Polish spaces such that $f(X)$ is uncountable, then there is a Cantor ...
TopologicalDynamitard's user avatar
3 votes
1 answer
94 views

Covering with sets of negligible boundary

I am studying causality theory in Lorentzian length spaces, and I have a question about geometric measure theory in general (it will help me with a proof I am trying to finish): Suppose we have a ...
Mathemachicken's user avatar
1 vote
1 answer
114 views

Why is $A^\mathbb{N}$ with the discrete topology a polish space

I am currently preparing for a part of a seminar in topology/descriptive set theory and am working with the A. Kechris' book. I am confused about some results for one of the easier examples, the ...
Taleofwoe's user avatar
1 vote
0 answers
131 views

Convergence in distribution (with measurable functions)

The definition of convergence in distribution: Let $S$ be a seperable metric space and $(X_n)_{n\in\mathbb{N}}$ a $S$-valued sequence of random variables. Then $(X_n)_{n\in\mathbb{N}}$ converge in ...
toni_iva's user avatar
  • 113
6 votes
0 answers
191 views

Markov Kernels Corresponding to Conditional Probabilities.

Let $(\Omega,\mathcal A,P)$ be a probability space and $\mathcal F$ a sub-$\sigma$-algebra of $\mathcal A$. Then the map $A\mapsto P[A|\mathcal F]$ is not a probability measure on $(\Omega,\mathcal A)$...
Alphie's user avatar
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2 votes
1 answer
82 views

Preimage measurable w.r.t. Effros Borel structure

For a Polish space $X$ let $F(X)$ the set of all closed subsets of $X$. The space $F(X)$ can be equipped with the $\sigma$-algebra generated by the sets of type $\{F \in F(X) : F \cap U \neq \...
user940347's user avatar
0 votes
1 answer
165 views

Total variation and probability measures

I'm currently reading some notes about optimal transport and here there is the definition of total variation:$$\Vert\mu\Vert_{TV}=2\sup\vert\mu(A)\vert$$where $\mu$ is a probability measure of the ...
Mathemachicken's user avatar
1 vote
1 answer
361 views

Suslin sets and projections

In a paper I am reading about optimal transport (we are in a proper metric space, https://arxiv.org/pdf/2004.08934.pdf, page 49 at the end), there's written in a proof "Being (this set) the ...
Mathemachicken's user avatar
2 votes
0 answers
207 views

Projection of an Analytic Set is Analytic

Let $\mathscr{I}$ be the family of all finite sequences in $\mathbb{N}$ and let us use Greek letters $\alpha,\beta\ldots$ to denote elements of $\mathscr{I}$. Given a paved space $(X,\mathscr{X})$, a ...
Dilemian's user avatar
  • 1,045
1 vote
1 answer
67 views

$(X, \mathcal{T})$ Polish and $F \subseteq X$ closed $\implies$ topology generated by $\mathcal{T} \cup \{ F \}$ is Polish

I am trying to understand the proof of this statement (Kechris, Classical Descriptive Set Theory, p. 82). (13.2) Lemma. Let $(X, \mathcal{T})$ be Polish and $F \subseteq X$ closed. Let $\mathcal{T}_F$...
qwertz's user avatar
  • 219
2 votes
1 answer
90 views

Second-countable, analytic, completely Baire spaces

First a few definitions. A topological space $(X,\tau)$ is said to be: a Polish space if it is separable and completely metrizable. Analytic, if there exists a surjective continuous map from a (Borel ...
Lorenzo's user avatar
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-1 votes
1 answer
105 views

Uncoutable Borel Set of Polish Space Contains Cantor Set: Proof Step

Currently, I am struggling to fully understand the following theorem from Alexander Kechris' book “Classical Descriptive Set Theory”: The last sentence is unclear to me. Why is the homeomorphic copy ...
qwertz's user avatar
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