Questions tagged [borel-sets]
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108 questions
3
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1
answer
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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{...
- 33
0
votes
1
answer
164
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/...
- 137
14
votes
1
answer
735
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 (...
- 345
4
votes
0
answers
64
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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
6
votes
3
answers
1k
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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
0
votes
1
answer
282
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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
1
vote
2
answers
266
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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\{\...
- 5,407
0
votes
1
answer
136
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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
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
3
votes
1
answer
619
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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
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
2
votes
0
answers
858
views
Regularity of locally finite Borel measure
Do you know any proof that locally finite Borel measure on metric space is regular ? I found many proofs only for finite Borel measure, but it's not satisfies me. Or maybe do you know any books or ...
- 143
5
votes
1
answer
310
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Abstract transverse measure theory
After reading Noncommutative Geometry book (see here) I came across the notion of the so called abstract transverse measure theory which is a generalization of standard measure theory well adapted to ...
- 9,330
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
2
votes
2
answers
332
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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
8
votes
1
answer
230
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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
$$
\mathscr{H}=\{\...
- 1,697
0
votes
1
answer
645
views
Total variation norm estimate
I have the following question concerning an estimate of the total variation norm. Let $\mu$ be a bounded Borel measure on $\mathbb{R}$ and denote by $\mu_t$ the measure defined by $\mu_t(\Omega):=\mu(\...
16
votes
2
answers
1k
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Is it known how the Sigma Algebra generated by Jordan measurable sets compares to universally measurable sets and analytic sets?
Unlike the collection $L$ of Lebesgue measurable sets, the collection $J$ of Jordan measurable sets do not form a Sigma algebra. (A set is Jordan measurable if and only if its characteristic function ...
- 4,587
3
votes
1
answer
735
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Borel $\sigma$-algebra in $\beta \mathbb N \times \beta \mathbb N$
For a second-countable space $X$, we have $${\rm Bor}(X\times X) = {\rm Bor}\, X \otimes {\rm Bor}\, X,$$ that is the Borel $\sigma$-algebra of the product is the product $\sigma$-algebra. Some ...
- 105
5
votes
3
answers
1k
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Regular Borel measures and the measure of a singleton
I'm studying this paper: http://matwbn.icm.edu.pl/ksiazki/sm/sm73/sm7313.pdf
At the top of page 36, it states the following Proposition:
Let $S$ be a compact and $\mu$ a regular Borel measure on $...
- 563
3
votes
0
answers
29
views
Borel rank of certain automorphism orbits in $L_p$ lattices
For any $p$ with $1\leq p < \infty$, let $L_p([0,1])$ be the Banach lattice of $L_p$ functions on the unit interval (with the standard measure). Let $A=\{f\in L_p([0,1]):\left\lVert f \right \...
- 12.4k
1
vote
1
answer
51
views
Complexity of set of fibers on which a set is relatively clopen
Let $X$ and $Y$ be compact metrizable spaces with $f:Y\rightarrow X$ an open surjection. Suppose that $G\subseteq Y$ is a closed set. How topologically complicated can the set $\{x\in X : f^{-1}(x)\...
- 12.4k
3
votes
1
answer
568
views
Does there exist a Lebesgue nonmeasurable set $E$ in $\mathbb{R}$ satisfies that $E\cap A$ is a Borel null set for every Borel null set $A$?
Let $\mathcal{B}_{\mathbb{R}}$ be the Borel $\sigma$-algebra on $\mathbb{R}$ and $\mu_L$ be the Lebesgue measure on $\mathbb{R}$.
Define a new $\sigma$-algebra $\mathcal{B}_0$ as follows:
$$\mathcal{...
- 1,997
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
10
votes
0
answers
161
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Borel sets with almost equal sections
This is a corrected version of a Q posted yesterday.
Suppose that $B\ne\varnothing$ is a planar lightface $\varDelta^1_1$ set, such that all its vertical cross-sections $B_x$, $x\in\text{proj}\,B$, ...
- 2,281
1
vote
1
answer
245
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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
14
votes
2
answers
539
views
Non-isomorphic measurable spaces
Suppose that $X$ and $Y$ are measurable spaces with the property: there are measurable bijections $f:X \to Y$ and $g:Y \to X$. Is it possible to find non-isomorphic spaces $X,Y$ with this property? ...
- 9,330
6
votes
1
answer
394
views
Set of integral curves of a vector field
Let $V \colon [0,1]\times \mathbb R^d \to \mathbb R^d$ be a Borel vector field which is globally bounded, $V \in L^\infty$.
I am looking for a reference for the following result (which I suppose it ...
- 980
1
vote
0
answers
188
views
Regular measure in finite Borel sets [closed]
I have a question concerning
these lecture notes, https://www.math.leidenuniv.nl/~vangaans/jancol1.pdf
In the proof of the proposition 2.3 (page 3), there are two steps:
1) define the family $\...
- 111
2
votes
2
answers
233
views
Is the domain of symmetric derivative borel set?
Let $\mu$ be the $n$-dimensional Lebesgue measure and $\lambda$ be a complex Borel measure on $\mathbb{R}^n$.
Let $S$ be the set of points $x\in \mathbb{R}^n$ where $\lim_{r\to 0} \frac{\lambda (B(x,...
- 337
4
votes
0
answers
414
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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
1
vote
1
answer
321
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A question about Borel sets on the unit interval
It is known that each non-decreasing continuous function $\phi$ induces a $\sigma$-additive measure $d\phi$ such that $\int_0^1 f(x) d\phi(x)$ exists for every bounded real-valued Baire function $f$. ...
- 5,902
0
votes
1
answer
178
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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
6
votes
0
answers
216
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Analytic equivalence relations whose classes are sometimes Borel
There are analytic equivalence relations for which the statement "All classes are Borel" is independent of $ZFC$. In all the examples I know about, the classes are non Borel in $L$ or $L[z]$ for some ...
- 61
6
votes
1
answer
772
views
When do Borel $\sigma$-algebras generated by the total variation norm and the weak* topology coincide?
I am almost certain that I read somewhere that the following is true, but I cannot seem to locate the reference. I would be most appreciative if someone could point me to a reference. The result was ...
user12713
3
votes
2
answers
700
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Is a sigma-finite Borel measure over $\mathbb R$ determined by its values on the continuous functions? [closed]
Suppose that $\mu$ and $\nu$ are sigma-finite measures on the Borel sigma-algebra over $\mathbb R$ such that $\int_{\mathbb R}f\,d\mu=\int_{\mathbb R}f\,d\nu$ for all nonnegative continuous functions $...
- 128k
6
votes
3
answers
1k
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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
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
4
votes
3
answers
688
views
Does every separated measurable space embed into a power of $\{0,1\}$?
Let $(X,\Sigma)$ be a measurable space of arbitrary cardinality. I would like to understand under which conditions this space is isomorphic to a measurable subset of $\{0,1\}^\kappa$ for some cardinal ...
- 6,406
3
votes
1
answer
234
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Reference to definition of matrix log with domain SO(3) which is Borel measurable
I was trying to set up an inverse to matrix exponential $\exp:\mathrm{Skew}(3\times 3)\to SO(3)$ that "covers" the biggest possible domain and is Borel measurable. I was wondering if there is a ...
- 203
10
votes
1
answer
3k
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Radon-Nikodym derivatives as limits of ratios
Let $\mu_1$ and $\mu_2$ be measures with $\mu_1 \ll \mu_2$. Suppose we can characterize (a version of) their Radon-Nikodym derivative this way:
$$\frac{d\mu_1}{d\mu_2}(x) = \lim_{n \to \infty} \frac{\...
- 629
2
votes
1
answer
352
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Measurability of a 'cone'
Let A be a (Lebesgue) measurable set in $ \mathbb{R}^n$. Consider the 'cone with base A' $A(1) = \{\alpha x \in \mathbb{R}^n : x \in A, \alpha \in (0,1] \}$.
Is B Lebesgue measurable? I assume it is, ...
- 101
1
vote
1
answer
216
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Open set of geodesics implies the set of starting points is open
Let $X$ be a complete and separable metric space, let $G(X) \subset C([0,1],X)$ be the space of continuous curves from $[0,1]$ to $X$ with constant speed, i.e.
$$ d(f(t),f(s)) = |t-s| d(f(0), f(1)). $$...
- 41
-1
votes
1
answer
259
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Absolute continuity of probabilities on Polish spaces and open sets. [closed]
On a polish space $\mathcal{X}$ i consider two Borel probabilities $P$ and $Q$ such that for any open set $E$ of $\mathcal{X}$ we have : $P(E) =0$ implies $Q(E)=0$. Does this imply that $Q$ is ...
- 41
6
votes
3
answers
1k
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Is the set of the absolutely continuous functions a Borel set of the space of the continuous functions?
Does anyone knows whether the set of the absolutely continous functions $F :[0,1]\to \mathbb{R}^d$ of the form $$F(t)= a + \int_0^tf(s) ds$$ where $f$ is an integrable function is a Borel set of the ...
- 125
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
3
votes
0
answers
185
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Unbounded Class of Orbit Equivalence Relations
In their paper titled "The Classification of Hypersmooth Borel Equivalence Relations" Alexander Kechris and Alain Louveau quote the following (Theorem 5.2 in the article) as "Harrington, unpublished": ...
- 392
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
3
votes
1
answer
340
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Any subset of Baire space is a union of a boldface $\Delta_2^0$ set and a set with no isolated points. Anybody know how to prove this?
I'm trying to do due diligence and determine whether this is known, trivial, original, etc. I have a proof of:
Theorem: If $S\subseteq \mathbb{N}^{\mathbb{N}}$ then $S=X\cup Y$ for some $X$ which is ...
- 121
4
votes
0
answers
570
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Is this observation about the Borel Hierarchy trivial?
Hello, consider the following theorem. Is it trivial? Is it interesting? Is it worth including in a paper if I can prove it in 1 line as a corollary?
Theorem: Suppose $n>0$ is a natural. ...
- 121