Questions tagged [borel-sets]

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What is an "open Baire set"?

In Measures Which Agree on Balls by Hoffmann-Jørgensen, it is stated that if $\varphi$ is a Baire function (which I presume means a pointwise limit of continuous functions), then $\{a<\varphi\}$ is ...
i like math's user avatar
1 vote
1 answer
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Borel functions in C*-algebras

Is there a way of defining representations of separable $C^*$-algebras, say $\Phi$, so that $\Phi(A)$ is faithful representation of $A$ on a separable Hilbert space. There is a closure operation $A\...
user52345435's user avatar
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1 answer
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Borel sets in Vietoris topology

Let $\mathcal{K} = \mathcal{K}(\mathbb{N}^{\mathbb{N}})$ be the set of all non-empty compact subsets of the Baire space $\mathbb{N}^\mathbb{N}$ equipped with the Vietoris topology. Let $G$ be a Borel ...
Arkadi Predtetchinski's user avatar
1 vote
0 answers
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Borel structure/sets coming from strong operator topology vs norm topology

Let $X, Y$ be Banach spaces. Moreover, let $\mathcal{L}(X,Y)$ be the space of bounded linear operators equipped with the standard operator norm topology, and $\mathcal{L}_{\mathrm{s}}(X,Y)$ the same ...
Marek Kryspin's user avatar
1 vote
0 answers
138 views

Polish spaces and analytic sets

Can we conclude that an analytic subset $A$ of a Polish space $X$ is also Polish? Let $\mathcal{M}(R^d)$ denotes the family of Borel probability measures on $R^d$ equipped with the Lévy-Prokhorov ...
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3 answers
764 views

How to prove that the Lebesgue $\sigma$-Algebra is not countably generated?

How to prove that there can't exist a countable set $\{A_1,A_2,\dots\}\subset \mathcal{L}(\mathbb{R})$ (where $\mathcal{L}(\mathbb{R})$ denotes the family of all Lebesgue measurable sets) such that $\...
Joris Wk's user avatar
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Relative position of flags for the general linear group

This question is related my question in An example of a Deligne–Lusztig variety for a general linear group, and the obtained answer. Situation I am working with the general linear group. Specifically, ...
EJB's user avatar
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Relative position of flags and the Robinson-Schensted correspondence

This question is related my question in An example of a Deligne–Lusztig variety for a general linear group, and the obtained answer. I am currently reading Steinberg, Robert, An occurrence of the ...
EJB's user avatar
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1 answer
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An example of a Deligne–Lusztig variety for a general linear group

Take $G=\operatorname{GL}_3$, defined over the algebraic closure of a finite field $\mathbb{F}_q$ and let $X$ be the set of Borel subgroups of $G$. The Frobenius morphism $F:G\to G$ induces a map $F:...
EJB's user avatar
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2 answers
245 views

Product of locally Borel sets locally Borel

Let $X$ be a locally compact Hausdorff space with a fixed Radon measure (= Borel measure that is finite on compact subsets, inner regular on open subsets and outer regular on Borel sets) $\mu$ . A ...
Andromeda's user avatar
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How to characterize the Borel sets of product between finite and uncountable space?

Consider the product space $Z=X\times Y$, where $X$ is a finite set with discrete topology and $Y$ is an uncountable compact subset of $\mathbb{R}^n$ with the usual subspace topology. Denote with $\...
cha0skampf's user avatar
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2 answers
502 views

Can you fit a $G_\delta$ set between these two sets?

Every subset of $\mathbb N \times \mathbb N$ can be viewed as a relation on $\mathbb N$. The set $\mathcal P(\mathbb N \times \mathbb N)$ of all relations on $\mathbb N$ has a natural topology with ...
Will Brian's user avatar
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0 answers
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Can every Borel set be partitioned into $\leq\!\aleph_1$ $F_{\sigma \delta}$ sets?

Consider the following two facts, a modified version of which appear in this paper of Arnie Miller from the early 1980's: $\bullet$ If $\mathbb R$ can be partitioned into $\aleph_1$ closed sets, then ...
Will Brian's user avatar
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2 answers
347 views

Which topological spaces have a standard Borel $\sigma$-algebra?

Call a topological space $X$ standard Borel if $X$ is standard Borel as a measurable space (equipped with its Borel $\sigma$-algebra), i.e. if there is a Borel isomorphism between $X$ and a Polish ...
Antoine Labelle's user avatar
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1 answer
359 views

Wild classification problems and Borel reducibility

My question is whether the archetype of 'wild' problems in algebra, namely classifying pairs of square matrices up to similarity, is 'non-smooth' in the sense of Borel reducibility. This was ...
John Baez's user avatar
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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
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Boolean algebra of ambiguous Borel class

Suppose $X$, $Y$ are uncountable compact metric spaces and $\Delta^0_\xi(X)$, $\Delta^0_\xi(Y)$ ($2\le\xi\le\omega_1$) are the Boolean algebras of Borel sets of ambiguous class $\xi$. So for $\xi=2$ ...
Fred Dashiell's user avatar
6 votes
1 answer
223 views

Extending a finite Baire measure to a regular Borel measure

Let $X$ be a Hausdorff compact space, and let $\mathrm {Ba}$, $\mathrm {Bo}$ be its Baire, respectively, Borel, $\sigma$-algebras. Let $\mu:\mathrm {Ba}\to[0,+\infty)$ be a finite Baire measure: it is ...
Pietro Majer's user avatar
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Is the point-wise limit of simple functions a measurable function?

Let $X$ and $Y$ be topological spaces. By a simple function $\phi: X\to Y$ we mean a finite range Borel measurable function. Q. Is the point-wise limit of a sequence of simple functions a Borel ...
ABB's user avatar
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1 answer
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Borel sigma algebra coming from the weak topology on TVS

Let $(X,\tau)$ be a topological vector space. Suppose that, there is a sequence of subsets $X_n\subseteq X$ with, For every $n\in \mathbb{N}$, the topology $\tau$ on $X_n$ is second countable and ...
ABB's user avatar
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1 vote
1 answer
144 views

Find a Borel measure such that the closed sets aren't arbitrarily close to the Borel sets with finite measure

I would like example of measures which shows that the following propositions are false: Proposition 1: Let $\mathfrak{B}$ be the Borel $\sigma$-algebra of a topological space $X$ and $\mu:\mathfrak{B}...
rfloc's user avatar
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Every Borel linearly independent set has Borel linear hull (reference?)

I am looking for a reference to the following fact, which probably is known and could be proved somewhere by someone. Theorem. The linear hull of any linearly independent Borel set in a Polish ...
Taras Banakh's user avatar
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Separating two sets by a $\boldsymbol{\Delta}_3^0$ set

Let $X$ be a Polish space and $A,B\subseteq X$ be such that $A\cap B = \emptyset$, we know that if there is no $\boldsymbol{\Delta}_2^0$ set separating $A$ from $B$ then there exists a Cantor set $C\...
Lorenzo's user avatar
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2 votes
1 answer
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Is every Borel function a projection of a Borel function with closed graph?

Is it true the following statement? Given two Polish spaces $X,Y$ and a Borel function $f:X\rightarrow Y$, there exists a Polish space $Z$ and a Borel function $g:X \rightarrow Y\times Z$ with closed ...
Lorenzo's user avatar
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3 votes
0 answers
127 views

Is the singular value decomposition a measurable function?

$\DeclareMathOperator\svd{svd}$Consider the SVD of rectangular matrices as operators $$\svd:\mathbb C^{n\times m}\to \mathbb U_n\times \mathbb D_{n, m}\times\mathbb U_m$$ where $\mathbb U_n$ is the ...
Exodd's user avatar
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9 votes
1 answer
388 views

VC dimension of Borel sets [duplicate]

Can there be an uncountable set $S\subseteq\mathbb R$ such that for each subset $D\subseteq S$, there is a Borel set $U$ with $D=S\cap U$? I'm asking merely out of curiosity, but I'll mention that ...
Bjørn Kjos-Hanssen's user avatar
6 votes
1 answer
331 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
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1 answer
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I can't believe it's not Replacement!

(I feel like I might have to apologise in advance for this question, but oh well..) I just rediscovered a comment from Asaf K here on MO that states that full Replacement is not needed for Borel ...
David Roberts's user avatar
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1 answer
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$f_\epsilon=\inf\{f(y):|y-x|<\epsilon\}$ is measurable Borel [closed]

I found this problem I have tried but it has been a bit complicated for me, Let $f:\mathbb{R}\to\mathbb{R}$ a bounded function. For each $\epsilon>0$, let $f_\epsilon (x)=\inf\{f(y):|y-x|<\...
Zaragosa's user avatar
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17 votes
1 answer
397 views

Partitions of the real line into Borel subsets

Problem 1. Is it true that for every cardinal $\kappa\le\mathfrak c$ there exists a partition $(B_\alpha)_{\alpha\in\kappa}$ of the real line into $\kappa$ pairwise disjoint non-empty Borel subsets? ...
Taras Banakh's user avatar
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3 votes
1 answer
179 views

A question on Borel equivalence relations

Suppose that $\mathsf E$ is a countable Borel equivalence relation on the reals, and $\mathsf B$ is a finer equivalence of order 2, so that each $\mathsf E$-class consists of precisely two $\mathsf B$-...
Vladimir Kanovei's user avatar
3 votes
1 answer
145 views

Can there be an upper bound on the Borel rank of the preimages of Borel sets under a surjective Borel map?

Let $X$ and $Y$ be standard Borel spaces, $Y$ uncountable, and $f : X \to Y$ a surjective Borel map. Is it possible that there is a countable ordinal $\alpha$ such that for each Borel set $B \subseteq ...
Arkadi Predtetchinski's user avatar
4 votes
1 answer
690 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
  • 1,602
6 votes
1 answer
199 views

Topological complexity of ultrafilters in $2^\kappa$ for uncountable $\kappa$

It is a well known fact that if $\mathcal{F}$ is a non-principal ultrafilter on $\omega$, then the set $\{ \alpha \in 2^\omega : \alpha \in \mathcal{F}\}$ (conflating binary strings with subsets of $\...
James Hanson's user avatar
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15 votes
2 answers
488 views

Nontrivial signed measure on Lebesgue measurable sets being trivial on Borel sets

Let $\mathfrak{L}(\mathbb{R})$ be the collection of Lebesgue measurable sets and $\mathfrak{B}(\mathbb{R})$ be the Borel sets. Question: Is there a nontrivial signed measure on $\mathfrak{L}(\mathbb{R}...
Zhang Yuhan's user avatar
6 votes
1 answer
279 views

A rather non-$F_\sigma$ Borel set

I asked this question at MSE a week ago, but received no answer, so I cross-post it here. I obtained a negative answer to this MSE question provided each metric space $X$ such that $|X|=\frak c$ and ...
Alex Ravsky's user avatar
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0 votes
0 answers
366 views

Weak topology on spaces of measures and Borel sets

Let $K$ be a compact Hausdorff space (not necessarily metric or even separable). Let $M(K)$ be the space of all Radon measures on $K$ (that is, finite signed regular Borel measures) endowed with the ...
Damian Sobota's user avatar
4 votes
0 answers
266 views

Sierpinski's characterization of $F_{\sigma\delta}$ spaces

According to [2]: Let $X$ be a space. We call a system $(X_s)_{s\in T}$ a Sierpinski stratification of $X$ if $T$ is a nonempty tree over a countable alphabet and $X_s$ is a closed subset of $X$ for ...
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5 votes
0 answers
337 views

Computing the infinite dimensional Lebesgue measure of "cubes"

There is no Lebesgue measure in infinite dimensions—this slogan is familiar to every student interested in analysis. One possible, precise statement of this result may be as follows: if $X$ is an ...
truebaran's user avatar
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2 votes
1 answer
175 views

The Borel class of a subset of $\mathbb Z^\omega$

Define $F(t)=\ln(t+1)$ for $t\geq 0$. For each sequence of integers $ s=s_0s_1s_2...\in \mathbb Z^\omega$ define $$t^*_{ s}=\sup_{n\geq 0}F^{n}(|s_n|)$$ where $F^{n}$ is the $n$-fold composition of $F$...
D.S. Lipham's user avatar
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-3 votes
2 answers
155 views

Getting almost certainty from uncountably many low-probability events

Let $(\Omega,\Sigma,\mathbb{P})$ be a complete probability space, $B\subseteq X$ be a non-empty Borel subset of a polish space $X$, $A$ be an uncountable indexing set, and $\{X_{\alpha,n}\}_{a \in A, ...
ABIM's user avatar
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-1 votes
1 answer
69 views

Maximum of bounded expectations at a certain Borel set?

Assume ${\bf x} \in \mathbb{R}^n$ denotes a real-valued bounded random variable with a distribution measured on the Borel space $(\mathbb{R}^n,\mathcal{B}^n)$. Let $f:\mathbb{R}^n\to\mathbb{R}$ denote ...
Its_me's user avatar
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2 votes
0 answers
113 views

Borel measurability

Suppose we have two locally compact Hausdorff spaces $X$ and $Y$. Let $i:X\to Y$ be a continuous injection. Under what condition the Borel $\sigma$-algebra of $X$ and $i(X)$ are isomorphic via the map ...
A beginner mathmatician's user avatar
3 votes
1 answer
319 views

Countable convergence-determining class for weak convergence of probability measures

Suppose that $E$ is a Polish space. Portmanteau theorem asserts that a sequence $(\mu_n)$ of Borel probability measures weakly converges to a Borel probability measure $\mu$ (shortly, $\mu_n\overset{...
user154110's user avatar
0 votes
1 answer
137 views

separable support of Borel measure, with tau-additive measure and full support

I have a problem with Proposition 7.2.10 in Bogachev's Measure Theory Volume II book on page 77 (I have link to my drive with that book https://drive.google.com/file/d/...
elsnar's user avatar
  • 127
14 votes
1 answer
607 views

Does there exist a non-zero signed finite borel measure which is zero on all balls?

Let $(X,d)$ be a compact separable metric space. Let $\mu$ be a Borel, regular, finite, signed measure on $X$ such that for all $x\in X$, for all $r>0$, $\mu(B(x,r))=0$, where $B$ denotes the (...
tisydi's user avatar
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4 votes
0 answers
63 views

Borel rank collapse in Hilbert cube modulo $\sigma$-ideal generated by zero-dimensional sets

Both of the commonly studied $\sigma$-ideals (meager sets and null sets) in Polish spaces with a natural measure (i.e. $\mathbb{R}$, $[0,1]$, $[0,1]^\omega$, $2^{\omega}$, etc.) have the nice property ...
James Hanson's user avatar
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3 votes
2 answers
773 views

Disjoint union of measures

This is a sort of follow-up question to this old post I came across. Setup: Let $\{X_n\}_{n \in \mathbb{N}}$ be a collection of Hausdorff topological spaces and let $\{\Sigma_n\}_{n \in \mathbb{N}}$ ...
ABIM's user avatar
  • 4,989
0 votes
1 answer
272 views

Explicit examples of (probability) measures on $\prod \mathbb{R}$

Let $\prod_{n \in \mathbb{N}} \mathbb{R}$ be equipped with the Tikhonov product of the Euclidean topologies on $\mathbb{R}$ and let $B$ the corresponding Borel $\sigma$-algebra. What is are some ...
ABIM's user avatar
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1 vote
2 answers
243 views

Approximation of $\sigma$-finite Borel measures by equivalent finite measures

Let $(\mathbb{R}^d,\mathcal{B}(\mathbb{R}^d),\mu)$ be a $\sigma$-finite Borel measure on $d$-dimensional Euclidean space. Can one always construct a sequence of finite equivalent measures $\left\{\...
ABIM's user avatar
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