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3 votes
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Can we generalize the Kuratowski Extension Theorem to Souslin spaces?

The Kuratowski Extension Theorem says: Let $(X,\mathcal{A})$ be a measurable space, $Y$ be a polish space, $A\subseteq X$, and $f:A\to Y$ be a measurable map. Then there is a measurable function $F:X\...
rfloc's user avatar
  • 627
2 votes
1 answer
370 views

Relationship between Baire sigma algebra and Borel sigma algebra of an uncountable product

I've been trying to understand various questions to do with sigma algebras on uncountable product spaces. Let $T$ be an uncountable set and for each $t \in T$, let $\Omega_t$ be a topological space. ...
SBK's user avatar
  • 1,179
4 votes
1 answer
166 views

Is the set of clopen subsets Borel in the Effros Borel space?

Let $X$ be a Polish space and $\mathcal{F}(X)$ the set of closed subsets of $X$ endowed with the Effros Borel structure, generated by sets of the form $\{F\in \mathcal{F}(X):F\cap U\neq \emptyset\}$, ...
Iian Smythe's user avatar
  • 3,115
2 votes
0 answers
49 views

$\sigma$-compactness of probability measures under a refined topology

Denote Polish spaces $(X, \tau_x)$ and $(Y, \tau_y)$, where $X$ and $Y$ are closed subsets of $\mathbb{R}$. Consider a Borel measurable function $f: (X \times Y, \tau_x \times \tau_y) \rightarrow \...
Hans's user avatar
  • 195
2 votes
1 answer
852 views

The Borel sigma-algebra of a product of two topological spaces

The following problem arose while trying to justify some "known results" in abstract harmonic analysis on noncommutative groups, for which I couldn't find explicit statements in the ...
Yemon Choi's user avatar
  • 25.8k
3 votes
1 answer
132 views

Does there exist $f:\Bbb{R}\to \Bbb{R}$ additive onto function such that $f(F) \subset \Bbb{R}$ has the property of Baire for every $F$?

Let $F\subset \Bbb{R}$ intersect every closed uncountable subsets of $\Bbb{R}$. Does there exist $f:\Bbb{R}\to \Bbb{R}$ additive onto function such that $f(F) \subset \Bbb{R}$ has the property of ...
SoG's user avatar
  • 307
1 vote
0 answers
155 views

Study of the class of functions satisfying null-IVP

$\mathcal{N}_u$ : Class of all uncountable Lebesgue-null set i.e all uncountable sets having Lebesgue outer measure $0$. Let $f:\Bbb{R}\to \Bbb{R}$ be a function with the following property : $\...
SoG's user avatar
  • 307
5 votes
1 answer
254 views

Is the topology of weak+Hausdorff convergence Polish?

Let $X$ be a compact metric space, $P_X$ the set of Borel probability measures on $X$, and $K_X$ the set of non-empty closed subsets of $X$. I will define the "topology of weak+Hausdorff ...
Julian Newman's user avatar
2 votes
1 answer
148 views

Borel $\sigma$-algebras on paths of bounded variation

Let $(C, \|\cdot\|)$ be the Banach space of continuous paths $x: [0,1]\rightarrow\mathbb{R}^d$ starting at zero with sup-norm $\|\cdot\|$. Let further $B\subset C$ be the subspace of $0$-started ...
fsp-b's user avatar
  • 463
1 vote
1 answer
183 views

Topological analog of the Lusin-N property

$A\subset \Bbb{R}$ is meager if $A$ can be expressed as a countable union of nowhere dense sets. Let $f:[a, b]\to \Bbb{R}$ is absolutely continuous, i.e., for every $\epsilon>0$, there exists $\...
SoG's user avatar
  • 307
10 votes
0 answers
272 views

What is the smallest $\sigma$-algebra of reals that is closed under addition of sets?

What is the smallest $\sigma$-algebra $\Sigma\subseteq\mathcal P(\Bbb R)$ containing the open sets and such that if $A,B\in\Sigma$, then $$A+B=\{a+b\mid a\in A,b\in B\}\in\Sigma?$$ I know that neither ...
Alessandro Codenotti's user avatar
6 votes
1 answer
353 views

A strong Borel selection theorem for equivalence relations

In Kechris' book "Classical Descriptive Set Theory" there is the following theorem (12.16): Let $X$ be a Polish space and $E$ an equivalence relation such that every equivalence class is ...
Daniel W.'s user avatar
  • 365
1 vote
0 answers
155 views

$f:Y\to X$ continuous with $f^{-1}(x)$ compact for $x\in X$, does there exist a Borel measurable map $g:X\to Y$?

Let $X,Y$ be Polish, metric spaces. $f:Y\to X$ is a continuous, surjective map and for any $x\in X$, $f^{-1}(x)\subset Y$ is compact. Is it true that there is a injective, Borel measurable map $g:X \...
mathmetricgeometry's user avatar
4 votes
1 answer
718 views

Is every element of $\omega_1$ the rank of some Borel set?

It is well known that we can obtain the $\sigma$-algebra of Borel subsets of $2^{\omega}$ in the following way: Let $B_0$ be the collection of all open subsets of $2^{\omega}$. For $\alpha=\beta+1$, ...
Hannes Jakob's user avatar
  • 1,799
3 votes
0 answers
92 views

Is there a T3½ category analogue of the density topology?

Motivation: I understand that various attempts have been made at defining a topology on $\mathbb{R}$ that is an analogue of the density topology ([1]) but for category (and meager sets) instead of ...
Gro-Tsen's user avatar
  • 32.5k
5 votes
1 answer
528 views

Base zero-dimensional spaces

Definition. A zero-dimensional topological space $X$ is called base zero-dimensional if for any base $\mathcal B$ of the topology that consists of closed-and-open sets in $X$, any open cover $\mathcal ...
Taras Banakh's user avatar
  • 41.8k
4 votes
0 answers
122 views

Completely I-non-measurable unions in Polish spaces

Problem. Let $X$ be a Polish space, $\mathcal I$ be a $\sigma$-ideal with Borel base, and $\mathcal A\subset\mathcal I$ be a point-finite cover of $X$. Is it true that $\mathcal A$ conatins a ...
Lviv Scottish Book's user avatar
4 votes
0 answers
195 views

A kind of 0-1 law?

Suppose that P is a Borel subset of Baire $\times$ Baire, such that for every pair $x,x'$ of reals in the horizontal copy of Baire, if: $x,x'$ are $E_0$-equivalent (that is, $x(n)=y(n)$ for all but ...
Vladimir Kanovei's user avatar
6 votes
0 answers
180 views

The Fubini property of the $\sigma$-ideal generated by closed subsets of measure zero

Question. Can a Borel subset $B\subset\mathbb R\times \mathbb R$ be covered by countably many closed sets of measure zero in the plane if for every $(a,b)\in\mathbb R\times\mathbb R$ the horisontal ...
Taras Banakh's user avatar
  • 41.8k
2 votes
0 answers
102 views

Is this concrete set generically Haar-null?

This question is related to this problem on MO about generically Haar-null sets in locally compact Polish groups but is more concrete. First we recall the definition of a generically Haar-null set in ...
Taras Banakh's user avatar
  • 41.8k
5 votes
0 answers
214 views

On generically Haar-null sets in the real line

First some definitions. For a Polish space $X$ by $P(X)$ we denote the space of all $\sigma$-additive Borel probability measures on $X$. The space $P(X)$ carries a Polish topology generated by the ...
Taras Banakh's user avatar
  • 41.8k
5 votes
0 answers
138 views

Disjoint covering number of an ideal

Let $\mathcal I$ be a $\sigma$-ideal with Borel base on an uncountable Polish space $X=\bigcup\mathcal I$. Let $\mathrm{cov}(\mathcal I)$ (resp. $\mathrm{cov}_\sqcup(\mathcal I)$) be the smallest ...
Taras Banakh's user avatar
  • 41.8k
12 votes
1 answer
316 views

A reference to a theorem on the equivalence of ideals of measure zero in the Cantor cube

I am looking for a reference of the following (true) fact: Theorem. For any two continuous strictly positive Borel probability measures $\mu,\lambda$ on the Cantor cube $2^\omega$ there exists a ...
Taras Banakh's user avatar
  • 41.8k
8 votes
0 answers
463 views

When is the sigma-algebra generated by closed convex sets the same as the Borel sigma-algebra

For which topological vector spaces $E$ do we have the equality between the sigma algebra generated by the closed convex subsets of $E$ and the Borel sigma algebra of $E$ ? More precisely, do we have ...
LCO's user avatar
  • 506
19 votes
1 answer
465 views

Large Borel antichains in the Cantor cube?

Let $2^\omega$ be the Cantor cube $\{0,1\}^\omega$, endowed with the standard compact metrizable topology and the standard product measure, called the Haar measure. The Cantor cube is considered as a ...
Taras Banakh's user avatar
  • 41.8k
13 votes
3 answers
820 views

Is there a Borel subset of $ \mathbb{R}^{2} $, with finite vertical cross-sections, whose projection onto the first component is non-Borel?

This question is related to another one that I asked two days ago. Question. Does there exist a Borel subset $ M $ of $ \mathbb{R}^{2} $ with the following two properties? The ...
Transcendental's user avatar
7 votes
1 answer
459 views

Product of limit $\sigma$-algebras

Let $X$ and $Y$ be Polish (i.e. Borel subsets of separable completely metrizable) spaces. For a Polish space $Z$, let $\mathscr{S}(Z)$ denote the limit $\sigma$-algebra on $Z$, i.e. the smallest $\...
Jefferson Huang's user avatar
-1 votes
1 answer
148 views

Continuity of function mapping $\mathcal{P}(\mathcal{P}(X))$ to $\mathcal{P}(X)$ [closed]

Given a topological space $Y$, let $\mathcal{P}(Y)$ be the set of all probability measures on $Y$, endowed with the weak* topology. Let $X$ be a topological space (for convenience, it might be Polish ...
user66910's user avatar
3 votes
1 answer
232 views

Is every set of small measure contained in an open set of small measure with null boundary?

Let $\lambda( \cdot )$ denote Lebesgue measure on $[0,1]$. Let $(A_n)_{n=1}^\infty$ be a decreasing sequence of Borel subsets of $[0,1]$ such that $\bigcap_{n=1}^\infty A_n = \emptyset$. Given $\...
burtonpeterj's user avatar
  • 1,769
4 votes
1 answer
1k views

Quotients of standard Borel spaces

Let $X$ and $Y$ be standard Borel spaces: topological spaces homeomorphic to Borel subsets of complete metric spaces. Given a surjective Borel map $f:X\to Y$, we get an equivalence relation $\sim_f\...
SBF's user avatar
  • 1,655
16 votes
4 answers
1k views

Continuity on a measure one set versus measure one set of points of continuity

In short: If $f$ is continuous on a measure one set, is there a function $g=f$ a.e. such that a.e. point is a point of continuity of $g$? Now more carefully, with some notation: Suppose $(X, d_X)$ ...
Nate Ackerman's user avatar
3 votes
1 answer
228 views

Product of Topological Measure Spaces

Def. A Radon measure $\mu$ on a compact Hausdorff space $X$ is uniformly regular if there is a countable family $\mathcal{A}$ of compact $G_\delta$-subsets of $X$ such that for every open set $U\...
Ameen's user avatar
  • 103
11 votes
1 answer
799 views

Restrictions of null/meager ideal

Let I denote the null (resp. meager) ideal on reals. Is it consistent that for any pair of non null (resp. meager) sets A and B, there is a null (resp. meager) preserving bijection between A and B? In ...
Ashutosh's user avatar
  • 9,631
13 votes
1 answer
1k views

Is every $\sigma$-algebra of sets *abstractly* the Borel algebra of a topology on perhaps some other set?

Is every sigma-algebra the Borel algebra of a topology? inspires the present question which asks for less. Question: Given a $\sigma$-algebra $\mathcal A$ on a set $X$, does there exist a topology $\...
David Feldman's user avatar
92 votes
3 answers
14k views

Is every sigma-algebra the Borel algebra of a topology?

This question arises from the excellent question posed on math.SE by Salvo Tringali, namely, Correspondence between Borel algebras and topology. Since the question was not answered there after some ...
Joel David Hamkins's user avatar
3 votes
0 answers
277 views

For METRIZABLE spaces, do the Banach classes and Baire classes coincide?

In this paper: 'Borel structures for Function spaces' by Robert Aumann, http://projecteuclid.org/euclid.ijm/1255631584 Aumann claims that when X and Y are metric spaces (among other things), the ...
Mario Carrasco's user avatar