Questions tagged [borel-sets]
The borel-sets tag has no usage guidance.
108 questions
0
votes
0
answers
112
views
Are measures singular with respect to all representing measures in $\mathbb{D}^n$ always concentrated on null-sets?
Let $\mu$ and $\nu$ be two measures on the $\sigma$-Borel set $\mathcal{B}(\mathbb{D}^n)$.
We say that $\mu$ is a representing measure for some point $z \in \mathbb{D}^n$, if
$$\forall_{u \in A(\...
- 63
0
votes
1
answer
78
views
Intersection of sigma algebras generated by shifts
EDIT: Iosif's answer showed that my motivation for this question was mislead.
To keep this question interesting for a broader readership, let us forget about sequence spaces and tail algebras and ...
- 257
3
votes
0
answers
65
views
Borel complexity of the set of generic points for an invariant measure in a minimal system
I would like to know what are possible Borel complexities of the set of generic points for a minimal topological dynamical system. The only possible complexity for which we do not know if it is ...
- 1,658
0
votes
0
answers
38
views
Are measures singular with respect to all representing measures in $\mathbb{D}^n$ always concentrated on null-sets? Will it also be a Henkin measure?
Let $\mu$ and $\nu$ be two measures on the $\sigma$-Borel set $\mathcal{B}(\mathbb{D}^n)$.
We say that $\mu$ is a representing measure for some point $z \in \mathbb{D}^n$, if
$$\forall_{u \in A(\...
- 63
0
votes
0
answers
78
views
What properties does representing measure $\mu$ for $z\in\mathbb{D}^n$ has to satisfy so that $\nu=0$ is the only measure singular with respect to it?
Let $\mu$ and $\nu$ be two measures on the $\sigma$-Borel set $\mathcal{B}(\mathbb{D}^n)$.
We say that $\mu$ is a representing measure for some point $z \in \mathbb{D}^n$, if
$$\forall_{u \in A(\...
- 63
5
votes
0
answers
95
views
Is there an equivalent condition for Borel projections being Borel?
Let $X,Y$ be Polish spaces, and $B\subseteq X \times Y$ a Borel subset. The projection $B_X$ is not necessarily Borel in $X$. I have seen a few sufficient conditions for the projection to be Borel, ...
- 291
1
vote
0
answers
43
views
Definition of "interval of continuity" for function defined on sets
At the beginning of Chapter 8 of Kubilius's Probabilistic Methods in the Theory of Numbers, the author defines $Q=Q(E)$ to be a completely additive nonnegative function defined for all Borel subsets $...
- 12.8k
-1
votes
1
answer
132
views
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 ...
- 297
1
vote
1
answer
209
views
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\...
- 13
2
votes
1
answer
83
views
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 ...
1
vote
0
answers
96
views
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 ...
- 139
1
vote
0
answers
154
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 ...
- 39
10
votes
3
answers
1k
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 $\...
- 243
1
vote
0
answers
86
views
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, ...
- 153
2
votes
1
answer
135
views
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 ...
- 153
5
votes
1
answer
443
views
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:...
- 153
4
votes
2
answers
261
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 ...
- 175
1
vote
1
answer
120
views
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 $\...
- 11
9
votes
2
answers
540
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 ...
- 18.5k
8
votes
0
answers
196
views
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 ...
- 18.5k
4
votes
2
answers
452
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 ...
- 3,446
10
votes
1
answer
398
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 ...
- 22.2k
2
votes
1
answer
851
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 ...
- 25.8k
5
votes
1
answer
254
views
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$ ...
- 1,704
7
votes
1
answer
280
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 ...
- 60.5k
1
vote
1
answer
692
views
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 ...
- 4,058
0
votes
1
answer
230
views
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 ...
- 4,058
1
vote
1
answer
157
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}...
- 627
3
votes
0
answers
80
views
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 ...
- 41.8k
1
vote
0
answers
80
views
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\...
- 2,286
2
votes
1
answer
115
views
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 ...
- 2,286
3
votes
0
answers
142
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 ...
- 201
9
votes
1
answer
396
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 ...
- 24.8k
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 ...
- 365
5
votes
1
answer
1k
views
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 ...
- 35.4k
0
votes
1
answer
361
views
$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|<\...
- 143
18
votes
1
answer
425
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?
...
- 41.8k
3
votes
1
answer
182
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$-...
- 2,281
3
votes
1
answer
155
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 ...
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$, ...
- 1,799
6
votes
1
answer
205
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 $\...
- 12.4k
15
votes
2
answers
530
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}...
- 807
6
votes
1
answer
298
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 ...
- 5,409
1
vote
0
answers
399
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 ...
- 1,151
4
votes
0
answers
273
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 ...
- 3,317
5
votes
0
answers
363
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 ...
- 9,330
2
votes
1
answer
177
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 $F$...
- 3,317
-3
votes
2
answers
156
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, ...
- 5,407
-1
votes
1
answer
74
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 ...
- 113
2
votes
0
answers
115
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 ...
- 1,879