Questions tagged [borel-sets]
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113 questions
2 votes
1 answer
146 views
Understanding proof that any injective Borel measurable function between Polish spaces is a Borel isomorphism onto its range (from Kechris' book)
I am trying to understand the proof of the following Corollary 15.2 in the book Classical descriptive Set Theory by Kechris. Corollary: Let $X$, $Y$ be standard Borel spaces and $f:X\rightarrow Y$ be ...
- 177
0 votes
0 answers
89 views
Understanding the theorem from Kechris' book that Borel sets are mapped to Borel sets under injective functions
Theorem 15.1 in Classical Descriptive Set Theory by Kechris states: (i) Let $X, Y$ be Polish spaces and $f:X\rightarrow Y$ be continuous. If $A\subseteq X$ is Borel and $f|_A$ is injective, then $f(A)$...
- 177
12 votes
1 answer
619 views
Can a Borel set in the plane intersect every arc but contain none?
A set $S\subseteq \mathbb{R}^2$ which intersects every arc but contains none can be constructed using transfinite recursion (just well-order the arcs then ensure each contains a point in $S$ and $\...
- 303
2 votes
1 answer
151 views
Optimal complexity of Borel bijections between Polish spaces
Let $X$ and $Y$ be uncountable Polish spaces and $f:X\to Y$ a Borel bijection (which automatically has Borel inverse); such a map $f$ exists by Kuratowski's theorem. Given countable ordinals $1\leq \...
- 255
7 votes
0 answers
338 views
Locally compact, second countable, Hausdorff topology refining the right order topology
Let $A\subseteq \mathbb R$ be Borel. Is there a second countable, locally compact, Hausdorff topology on $A$ that is finer than the right order topology (i.e. the topology with base $\{x\in A:a< x\}...
- 175
0 votes
1 answer
102 views
Is every Borel set a Caccioppoli set after alterations of measure zero?
Let $E\subset\mathbb R^n$ be a Borel set. Does there always exist a Caccioppoli set $F$ such that $\mathcal L^n(E\Delta F)=0$ (where $\mathcal L^n$ is the $n$-dimensional Lebesgue measure of $\mathbb ...
- 41
0 votes
0 answers
160 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 $$\forall_{u \in A(\...
- 53
0 votes
1 answer
226 views
Intersection of sigma algebras generated by shifts
EDIT: Iosif's answer showed that my motivation for this question was mislead. To keep this question interesting for a broader readership, let us forget about sequence spaces and tail algebras and ...
- 269
3 votes
0 answers
113 views
Borel complexity of the set of generic points for an invariant measure in a minimal system
I would like to know what are possible Borel complexities of the set of generic points for a minimal topological dynamical system. The only possible complexity for which we do not know if it is ...
- 1,678
0 votes
0 answers
110 views
What properties does representing measure $\mu$ for $z\in\mathbb{D}^n$ has to satisfy so that $\nu=0$ is the only measure singular with respect to it?
Let $\mu$ and $\nu$ be two measures on the $\sigma$-Borel set $\mathcal{B}(\mathbb{D}^n)$. We say that $\mu$ is a representing measure for some point $z \in \mathbb{D}^n$, if $$\forall_{u \in A(\...
- 53
5 votes
0 answers
127 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, ...
- 301
1 vote
0 answers
67 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 $...
- 13.1k
-1 votes
1 answer
174 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 ...
- 307
1 vote
1 answer
261 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
132 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
123 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
193 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 $\...
- 253
1 vote
0 answers
190 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, ...
- 171
2 votes
1 answer
194 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 ...
- 171
6 votes
1 answer
582 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:...
- 171
4 votes
2 answers
314 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 ...
- 257
1 vote
1 answer
165 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
616 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 ...
- 20.1k
8 votes
0 answers
230 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 ...
- 20.1k
5 votes
2 answers
712 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 ...
- 4,097
13 votes
2 answers
625 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 ...
- 24.5k
4 votes
2 answers
1k 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 ...
- 26.4k
5 votes
1 answer
306 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 $\xi=2$ ...
- 1,852
7 votes
1 answer
363 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 ...
- 63.5k
1 vote
1 answer
848 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 ...
- 4,140
0 votes
1 answer
266 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 ...
- 4,140
1 vote
1 answer
192 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}...
- 763
3 votes
0 answers
96 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 ...
- 44.3k
1 vote
0 answers
97 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\...
- 2,308
2 votes
1 answer
145 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 ...
- 2,308
3 votes
0 answers
184 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
441 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 ...
- 25.3k
6 votes
1 answer
423 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 ...
- 385
6 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 ...
- 36.8k
0 votes
1 answer
410 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
19 votes
1 answer
496 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? ...
- 44.3k
3 votes
1 answer
233 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 $\mathsf B$-...
- 2,431
3 votes
1 answer
213 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
815 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 $\alpha=\beta+1$, ...
- 1,819
6 votes
1 answer
242 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 $\...
- 19.4k
15 votes
2 answers
656 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 $\mathfrak{L}(\mathbb{R}...
- 1,007
6 votes
1 answer
361 views
A rather non-$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 ...
- 6,004
1 vote
0 answers
478 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 ...
- 1,161
4 votes
0 answers
302 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 ...
- 3,691