# Questions tagged [borel-sets]

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61
questions

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votes

**1**answer

117 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 $F$...

**0**

votes

**2**answers

144 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, ...

**-1**

votes

**1**answer

58 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 ...

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92 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 ...

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votes

**1**answer

40 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{...

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votes

**1**answer

38 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/...

**11**

votes

**1**answer

156 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 (...

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48 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 ...

**4**

votes

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211 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 $\{\Sigma_n\}_{n \in \mathbb{N}}$ ...

**1**

vote

**1**answer

230 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 ...

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votes

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77 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\{\...

**0**

votes

**1**answer

118 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 $A=f(O)$?
$B$ ...

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votes

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97 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_{\...

**3**

votes

**1**answer

153 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**

votes

**1**answer

197 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 ...

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votes

**0**answers

164 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 ...

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votes

**1**answer

197 views

### 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 ...

**7**

votes

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313 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 ...

**2**

votes

**2**answers

188 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 ...

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votes

**1**answer

203 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}=\{\...

**0**

votes

**1**answer

364 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(\...

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votes

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701 views

### 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 ...

**3**

votes

**1**answer

259 views

### 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 ...

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votes

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348 views

### 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 $...

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votes

**0**answers

26 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 \...

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vote

**1**answer

46 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)\...

**3**

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**1**answer

236 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**

vote

**1**answer

221 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{...

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votes

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142 views

### 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 $B_x$, $x\in\text{proj}\,B$, ...

**1**

vote

**1**answer

158 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 = \{...

**13**

votes

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463 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? ...

**6**

votes

**1**answer

247 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 ...

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vote

**0**answers

134 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 $\...

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votes

**2**answers

156 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,...

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319 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 ...

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vote

**1**answer

197 views

### 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 $f$. ...

**0**

votes

**1**answer

157 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, ...

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200 views

### 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 ...

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votes

**1**answer

460 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 ...

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votes

**2**answers

461 views

### 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 $...

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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 ...

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votes

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213 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 ...

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votes

**3**answers

504 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 ...

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votes

**1**answer

173 views

### 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 ...

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votes

**1**answer

1k views

### 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{\...

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votes

**1**answer

219 views

### 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, ...

**1**

vote

**1**answer

184 views

### 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.
$$ d(f(t),f(s)) = |t-s| d(f(0), f(1)). $$...

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votes

**1**answer

226 views

### 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 ...

**5**

votes

**3**answers

972 views

### 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 ...

**10**

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**0**answers

620 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 ...