Questions tagged [polish-spaces]

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

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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. ...
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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 \...
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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 ...
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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 ...
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Transition kernels for Markov chains

A Markov chain with general state space $(S,\mathbf{S})$ is specified by an initial distribution $\mu_0$ on $(S,\mathbf{S})$ and a transition kernel $P:S\times \mathbf{S}\to [0,1]$, that is, $B\mapsto ...
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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, ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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Why is a space of probability measures, endowed with the weak topology on probability spaces, a Polish space?

Why is a space of probability measures, endowed with the weak topology on probability spaces, a Polish space? The question comes from my studying of random walks in random environments. Suppose I have ...
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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 ...
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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 ...
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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 $(\Omega,\mathcal A)$...
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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 \...
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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 ...
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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 ...
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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 ...
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$(X, \mathcal{T})$ 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 $\mathcal{T}_F$...
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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 ...
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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 ...
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Borel isomorphisms between perfect Polish spaces and their "rank"

Let $\mathcal{X, Y}$ be perfect Polish spaces. Define $\Sigma_1^0$ sets to be open sets, and for $\xi>1$, $\Sigma_\xi^0$ sets to be sets of the form $\bigcup_{n=1}^\infty P_n^c$ where $P_n$ is a $\...
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Why the space of Katětov functions with finite support is separable?

Let $(X, \rho)$ be a metric space. A function $f: X \rightarrow \mathbb R$ is called Katětov map if $$|f(x) - f(y)| \le \rho(x, y) \le f(x) + f(y) \quad \forall x, y \in X.$$ Denote the space of all ...
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$(a,b)$ is polish space with induced topology

From Topology Without tears: Prove that each discrete space and each of the spaces $[a,b],(a,b),(a,b],[a,b),(-\infty,a),(a,\infty)\ and\ \{a\}\ for\ a,b\in \mathbb{R}$ with its induced topology is ...
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2 votes
1 answer
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Locally finite vs. Borel measures on $\sigma$-compact Polish spaces

Let $E$ be a Polish space, and let $\mu$ be a measure on $E$. Define the following properties: $E$ is $\sigma$-compact if $E$ is the countable union of compact sets. $E$ is locally compact if every $...
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On gaps in the dimension of subspaces

Let $P\subseteq\Bbb N\cup\{\infty\}$, with $0\in P$. Does there exist a "nice" space $X_P$ such that $X_P$ has a subspace of dimension $n$ iff $n\in P$? By nice I mean nice enough that there'...
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Are Radon measures on Polish spaces $\sigma$-finite?

If $\Omega$ is a Polish space and $\mu$ is a Radon measure on $\Omega$ (i.e. an inner-regular Borel measure), is $\mu$ $\sigma$-finite? I know that Radon measures in general need not be $\sigma$-...
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7 votes
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On positive dimensional Polish spaces in which every compact set has empty interior

A standard characterization of the Baire space is that is the only nonempty, zero dimensional, Polish space in which every compact set has empty interior (up to homeomorphism of course). I'm ...
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5 votes
1 answer
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Lusin’s Theorem for Polish spaces with infinite Radon measure

I’m working on the following exercise in Klenke’s Probability Theory: A Comprehensive Course (Exercise 13.1.3), which asks us to prove the following generalization of Lusin’s Theorem: Let $\Omega$ be ...
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2 votes
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Polish group actions: if an orbit is non-meager in itself, it is a Baire space?

Assume that $G$ is a Polish group continuously acting on a Polish space $X$. Let $x \in X$ be a point such that $G \cdot x$, the orbit of $x$, is non-meager in its relative topology. I would like to ...
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3 votes
1 answer
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If $\tau$ and $\tau'$ define the same borel sets, then $\tau=\tau'$

I have a group $G$ with two topologies $\tau, \tau'$ on $G$ that makes it a Polish group (a completely metrizable and separable topological group). I need to show that if $\mathcal{B}(\tau)=\mathcal{B}...
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2 votes
1 answer
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Katetov-maps with same support

Let (X,d) be a metric space. f: X $\rightarrow \mathbb{R}$ ist called "Katetov map" iff $\forall x, y, \in X : |f(x)-f(y)| \leq d(x,y) \leq f(x) + f(y)$. The set of all Katetov-maps on X is ...
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3 votes
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Group of isometries of a Polish metric space acting on the space – the orbit of a point

Assume that $(X,d)$ is a separable complete metric space a denote by $G$ the group of all isometries of $(X,d)$. It is well-known that $G$ equipped with the pointwise convergence topology is a Polish ...
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4 votes
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Completeness of the Fell topology

Recall that the Fell topology $\tau_F$ is a topology on the hyperspace $F(X)$ of closed subsets of a Hausdorff space (maybe you can define it in a more general context, but I am interested in $\mathbb{...
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0 votes
1 answer
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Countable comeager set in an (uncountable) non perfect polish space

Is it possible to have an uncountable and non perfect Polish space with a countable comeager set? Furthermore, is it possible for this space to have a comeager collection of isolated points? I've been ...
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2 votes
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Conditions for continuity of the map $x \mapsto \mu_x$ (where $ \mu = \intop \mu_x d \mu (x)$ is the ergodic decomposition of $\mu$)

Let $(X,\mathcal{B},\mu,T)$ be a measure preserving system on a Borel space. Let $\{\mu_x\}_x$ be the conditional measures from the ergodic decomposition of $\mu$ (that is, $ \mu = \intop \mu_x d \mu ...
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1 vote
1 answer
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Weak limit of coupling is a coupling.

Let $X, Y$ be Polish spaces with probability measure $\mu, \nu$. Let $(\pi_n)$ be a sequence of couplings of ($\mu,\nu)$ that converges weakly to $\pi$. Show that $\pi$ is a coupling of $\mu,\nu$ too....
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3 votes
1 answer
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When is the continuous image of a measurable subset of a Polish space measurable?

In page two of three of this note: http://math.iisc.ac.in/~manju/MartBM/RaoSrivastava_borelisomorphism.pdf It is said in the proof of Proposition 2, and in the section (ii) that '$f$ is clearly bi-...
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Embedding of Cantor space proof issue

I think there is an issue in this proof, where $i(n)$ and $k(n)$ need to be redefined slightly as follows for the intersection of the nested open balls to be a singleton: $i(n) = \inf \{m > n : \...
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1 vote
1 answer
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Perfect Kernel analogue for NON $\sigma-$compact subsets of a Polish space

I'm looking at the proof of the following theorem by Hurewicz (7.10 in Kechris' Descriptive Set Theory): Suppose $X$ is a Polish Space. Then one of the following holds: $X$ is $K_\sigma$ (i.e, $\...
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1 vote
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Is there a discontinuous closed injection of the first Baire class?

Is it possible to find a discontinuous function $f\colon X\to X$ on a Polish space $X$ such that $f$ is injective, $f$ is closed (maps closed sets to closed sets), and $f$ is of Baire first class? ...
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1 vote
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Why can I conclude $Q_n \xrightarrow{\mathscr{D}} Q$?

Suppose $X$ and $Y$ are Polish spaces. Suppose $Z \subset X$ is a borelian subset of $X$. Let $\varphi : Z \to Y$ continuos and $(R_n)_{n \geq 1}$ probabilities on $X$ such that for all $n \geq 1$ $...
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Convergence in Polish Spaces

If I have a point $x \in S$ where $S$ is some Polish Space with metric $d$, a closed set $F \subset S$, and $inf\;\{d(x,f)|f\in F\}=0$, then why can we say that there exists $f' \in F$ such that $d(x,...
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2 votes
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Dense subsets in the hyperspace of compact sets

Let $X$ be compact and Polish (I'm thinking of $X=[0,1]$, but I guess maybe the same answer holds for every compact Polish space) and let $\mathbf{K}(X)$ be the hyperspace of compact subsets of $X$, ...
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3 votes
1 answer
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Connectedness and complexity in Polish spaces

I was wondering: How complex can connected subsets of Polish spaces be? Are there connected non-Borel subsets of a Polish space? Given $X$ Polish space (not totally disconnected), does it have proper ...
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5 votes
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Total disconnection and zero-dimension in Polish spaces

First of all Polish spaces are completely-metrizable, separable topological space and by zero-dimensional Polish space I mean that the Polish space has a (countable) basis made of clopen sets. It is ...
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4 votes
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Does proving that closed subset of Polish space is Polish require axiom of countable choice?

Let $C$ be a closed subset of polish space $P$. It is trivial that $C$ is also completely metrizable, but how do we prove that $C$ is separable? I came up with this method: We can prove that separable ...
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Why is a Polish space a standard measurable space?

A measurable space ($\Omega$,$\mathcal F$) is called a standard measurable space if it is Borel isomorphic to one of the following measurable spaces: $(\langle 1, n\rangle,\mathcal B(\langle 1, n\...
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How do I prove this given set is open in the space of probability measures?

Let $X$ be a separable metric space and let $Prob(X)$ be the set of Borel probability measures on $X$, equipped with the weak* topology. Here are the facts that I know: The collection $\{\lambda\in ...
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