All Questions
Tagged with borel-sets gn.general-topology
36 questions
16
votes
5
answers
9k
views
A G-delta-sigma that is not F-sigma?
A subset of $\mathbb{R}^n$ is
$G_\delta$ if it is the intersection
of countably many open sets
$F_\sigma$ if it is the union of countably many closed sets
$G_{\delta\sigma}$ if it is the union
of ...
- 1,981
10
votes
0
answers
744
views
Is the set of real-valued lower semi-continuous functions measurable in epigraph topology (= topology of Gamma convergence)?
Let LSC = LSC([0,1]) be the set of non-negative, lower semi-continuous functions on the unit interval which take values in $\mathbb{R}_+ \cup \{\infty\}$. We use epigraph topology on LSC, i.e. a ...
- 888
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
3
answers
846
views
A compactness property for Borel sets
Is the following generalised compactness property of Borel sets in a Polish space consistent with ZFC?
($*$) Let $\mathcal{B}$ be a family of $\aleph_1$-many Borel sets. If $\bigcap \mathcal{B} = \...
- 866
8
votes
1
answer
938
views
Filling $\mathbb{R}^3$ with skew lines
I would like to know if it is possible to fill $\mathbb{R}^3$ with lines with the
following two properties:
(1) Every point $x \in \mathbb{R}^3$ is contained in precisely one line.
(2) Every ...
- 151k
8
votes
1
answer
230
views
Measure support decomposition that "tends to infinity"
I would like to know the answers to the following two questions.
Let $S$ be a locally compact Hausdorff space, $\mu$ be a regular Borel measure with non-compact support $M$. Denote
$$
\mathscr{H}=\{\...
- 1,697
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
7
votes
2
answers
500
views
Do continuous maps factor through continuous surjections via Borel maps?
Let $f \colon X \twoheadrightarrow Y$ be a continuous surjection between compact Hausdorff spaces, and $g \colon \mathbb{R} \to Y$ a continuous function. Can you always find a Borel-measurable ...
- 3,937
6
votes
3
answers
1k
views
Borel cross section
It is known from metric space topology that a closed equivalence relation on a Polish space has either countably many or $\mathfrak{c}$ many equivalence classes.
A short elementary proof is given in ...
- 1,704
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
6
votes
3
answers
1k
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}}$ ...
- 5,407
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
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
5
votes
1
answer
583
views
The Borel class of a countable union of $G_\delta$-sets, which are absolute $F_{\sigma\delta}$
Problem. Assume that a metrizable separable space $X$ is the countable union $X=\bigcup_{n\in\omega}X_n$ of pairwise disjoint $G_\delta$-sets $X_n$ in $X$ such that each $X_n$ is an absolute $F_{\...
- 41.8k
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
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
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
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
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
4
votes
0
answers
64
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 ...
- 12.4k
4
votes
0
answers
414
views
Topology on the space of Borel measures
Let $ B $ be the set of all measures $ \phi $ of $ \mathbf{R}^{n} $ such that every open set is $ \phi $-measurable (sometimes these measures are called Borel measures). Note the measures in $ B $ are ...
- 541
3
votes
1
answer
601
views
Is every closed subset of finite measure contained in an open subset of finite measure?
Could someone will verify my statement: For every locally finite Borel measure on metric space and closed set $F$ with finite measure, there exists open set $U$ such that $F \subset U$ and $U$ has ...
- 143
3
votes
2
answers
236
views
Can a continuous surjection from a Hilbert cube to a segment behave bad wrt Lebesgue measures?
Suppose $\hat{I}=[0,1]^\mathbb{N}$ is a Hilbert cube and $I=[0,1]$.
Consider Lebesgue measures $m_1$ and $m_2$ on $\hat{I}$ and $I$ correspondingly. By Lebesgue measure on the Hilbert cube I mean the ...
- 173
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
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$$
where $\mathbb U_n$ is the ...
- 201
3
votes
1
answer
619
views
Subsets of reals which are both $F_{\sigma\delta}$ and $G_{\delta\sigma}$
Let $X\subseteq \mathbb R$ such that
$X$ is an $F_{\sigma\delta}$-set (in $\mathbb R$); and
$X$ is a $G_{\delta\sigma}$-set.
It is not necessarily true that $X$ must be $F_\sigma$ or $G_\delta$. A ...
- 3,317
2
votes
2
answers
332
views
The space of Borel function modulo comeager sets is Dedekind complete
Let $(X,\tau)$ be a topological space. Denote by $Bor(X)$ the space of Borel functions $f:X\to\mathbb{R}$ where we identify two functions whenever they agree on the complement of a meager set. We ...
- 267
2
votes
1
answer
851
views
The Borel sigma-algebra of a product of two topological spaces
The following problem arose while trying to justify some "known results" in abstract harmonic analysis on noncommutative groups, for which I couldn't find explicit statements in the ...
- 25.8k
2
votes
1
answer
177
views
The Borel class of a subset of $\mathbb Z^\omega$
Define $F(t)=\ln(t+1)$ for $t\geq 0$.
For each sequence of integers $ s=s_0s_1s_2...\in \mathbb Z^\omega$ define $$t^*_{ s}=\sup_{n\geq 0}F^{n}(|s_n|)$$ where $F^{n}$ is the $n$-fold composition of $F$...
- 3,317
2
votes
0
answers
115
views
Borel measurability
Suppose we have two locally compact Hausdorff spaces $X$ and $Y$. Let $i:X\to Y$ be a continuous injection. Under what condition the Borel $\sigma$-algebra of $X$ and $i(X)$ are isomorphic via the map ...
- 1,879
1
vote
1
answer
245
views
Definition of $F_{\sigma}$ sets in terms of $\varepsilon$?
Let $X$ be a metric space.
In Borel hierarchy, $\Sigma_{1}^0$ is the set of all open sets in $X$ while $\Pi_{1}^0$ is the set of all closed sets in $X.$ Then at next level, one has $\Sigma_{2}^0 = \{...
- 623
1
vote
1
answer
51
views
Complexity of set of fibers on which a set is relatively clopen
Let $X$ and $Y$ be compact metrizable spaces with $f:Y\rightarrow X$ an open surjection. Suppose that $G\subseteq Y$ is a closed set. How topologically complicated can the set $\{x\in X : f^{-1}(x)\...
- 12.4k
1
vote
1
answer
869
views
Borel $\sigma$-algebra on the space of Hölder continuous functions
Let
$(M,d)$ be a separable metric space
$E$ be a $\mathbb R$-Banach space
$\alpha\in(0,1]$
Moreover, let $$\left\|f\right\|_{C^{0+\alpha}(K,\:E)}:=\sup_{x\in K}\left\|f(x)\right\|_E+\sup_{\substack{...
- 167
0
votes
1
answer
178
views
Borel subsets of Polish groups
Suppose that I have a polish group $G$ and two subsets $A$ and $B$ of $G$ such that: $A$ is open in $G$ and $B$ is closed in $G,$ from this, can I conclude that $AB$ is a Borel subset of $G$? if not, ...
- 313
0
votes
1
answer
282
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 ...
- 5,407
0
votes
1
answer
136
views
A link between continuity and 0-borelian? [closed]
Is it true that :
1/ if $f$ real continuous and $O$ an open set then $f(O)$ is a 0-borelian?
2/ if $A$ a 0-borelian set then there exists $f$ real continuous and $O$ an open set with $A=f(O)$?
$B$ ...
- 4,074