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Questions tagged [borel-sets]

For questions about Borel sets. Please, add also other tags indicating the area, e.g., (measure-theory), (general-topology), (descriptive-set-theory), etc.

2
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0answers
30 views

Borel action of $GL_n(\Bbb{Q})$ on $S(\Bbb{Q}^n)$

I'm reading a paper on torsion-free abelian groups and I'm trying to work out all the details. I know that the space of all torsion-free abelian groups of rank $1\le r\le n$, where $n$ is a positive ...
4
votes
1answer
67 views

Does every uncountable Borel subset of $\mathbb R$ contains a perfect subset?

This question came from (London Mathematical Society Student Texts) Krzysztof Ciesielski-Set Theory for the Working Mathematician-Cambridge University Press. Chapter 6.2 Exercise 5. I have thought ...
0
votes
1answer
35 views

Continuous then measurable?

I have this proposition: Prop. Every continuous functions $f:\mathbb{R}^n \to \mathbb{R}$ is $\mathcal{B^n} - \mathcal{B}$ measurable. I assume here $\mathbb{R}, \mathbb{R}^n$ count with their ...
0
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0answers
26 views

measurable function/ Floor and ceiling functions [closed]

I would like to find out, if $ f: (\mathbb{R}, B(\mathbb{R})) \rightarrow (\mathbb{R}, B(\mathbb{R}))$is measurable: $$ f(x)=\left\{\begin{array}{ll} x \cdot [\frac{1}{x}], & x\ne 0 \\ ...
3
votes
1answer
83 views
+50

Exercise about sub-$\sigma$-algebra of $\mathcal{B}(\mathbb{R})$

Let $C=\{(-a, a): a \in \mathbb{R}\}$ and $F=\sigma(C)$. Prove that $F=\mathcal{B}(\mathbb{R})\cap\{A\subseteq\mathbb{R}: A=-A\}$. I don't have problems in proving $F\subseteq \mathcal{B}(\...
1
vote
0answers
28 views

Intuition of Standard and Analytic Borel space

An analytic Borel space is a countably generated Borel space which is the image of standard Borel space under a Borel mapping. A standard Borel space is the Borel space associated to a Polish space. ...
0
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0answers
16 views

What is the natural $\sigma$-algebra on the set of functions $\mathbb{R}^{D}$?

Let $D$ be a topological space with Borel $\sigma$-algebra $\mathcal{B}(D)$. Then what is the natural $\sigma$-algebra on the space of functions from $D$ to $\mathbb{R}$ $$ \mathbb{R}^{D}: = \{f \...
0
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1answer
40 views

Proof borel sets are measurable sets [closed]

Proof that : if $B$ is Borel Set, then $B$ measurable set. I know the definition of measureable sets, for all $A\subseteq \mathbb{R}$, $m^*(A)=m^*(A\cap E)+m^*(A\cap E^C)$, which $m*$ denote the ...
0
votes
1answer
34 views

Is there such a term as a “Borel measurable set”?

Not sure if this is the right place to post such a rookie question, but I'd appreciate some quick clarification. Is there such a term as a "Borel measurable set"? I've seen it used all over the place ...
1
vote
0answers
21 views

Building a Random Element From a Distribution

Suppose that $(X,\mathscr{B}(X))$ is a standard Borel space, and let $\pi$ be a Borel-probability measure on $X$. Is there a way to construct an $X$-valued random element, with distribution $\pi$?
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13 views

Quotient Borel structure

I'm reading "Ergodic theory and Semisimple Lie groups" by Zimmer and (are p. $10$) the author states that ($G$ is locally compact and second countable): Definition $(2.1.9)$ Let $S$ be a Borel $G$...
0
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0answers
71 views

Prove that $\lim_{m(I)\rightarrow 0}\frac{m(E\cap I)}{m(I)}=c$ for some $c$ and moving interval $I$.

Let $E$ be a set such that $m(E)>0$, $E \subset (0,1)$. Prove that there exists $c>0$ such that for some moving interval $I$, $$\lim_{m(I)\rightarrow 0}\frac{m(E\cap I)}{m(I)}=c$$ Proof: $m(E)=...
0
votes
1answer
33 views

Borel measurability of a set

Hey guys I have a question on the Borel measurability of this set $\{{(x,y):x∈E,0<y<f(x)}\}$ when $f$ is a continuous function defined in an open set. Can anyone help me out? I think the set is ...
0
votes
2answers
22 views

$[0,1)$ is in both $G_{\delta}$ and $F_{\sigma}$

I know $G_{\delta}$ is the complement of $F_{\sigma}$ and it can be proved easily by by using the countable intersection of sets is the complement of the countable union of the complements of the sets....
0
votes
1answer
74 views

What kind of numbers are inside a generating open interval of the Borel $\sigma$-algebra? [closed]

If it is enough to have all open intervals (a,b) with end points $a$ and $b$ belonging to the rational numbers, a < b, in order to generate a Borel $\sigma$-algebra on $\mathbb{R}$. Asked here: ...
0
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1answer
19 views

Show this set generates the Borel subset of the reals?

This question is in a past paper I'm doing: Let $\mathcal{I}$ be the collection of all intervals in $\mathbb{R}$ not containing zero. Show that $\mathcal{I}$ generates the $\sigma$-algebra of Borel ...
2
votes
2answers
103 views

Question about Kallenberg's proof of Doob-Dynkin Lemma

The following is the proof for the Doob Dynkin Lemma from Kallenberg's Foundations of Modern Probability. In the theorem, $(S,\mathscr{S})$ is assumed to be Borel, i.e. Borel isomorphic to a Borel ...
0
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0answers
18 views

Veryfy some properties of a given Borel set.

for a given $n\in\mathbb{N}_{>0}$ $$A_{n,\epsilon}=\bigcup_{\frac{p}{q}\in\mathbb{Q}}(\frac{p}{q}-\frac{\epsilon}{q^n},\frac{p}{q}+\frac{\epsilon}{q^n}), A_n=\bigcap_{\epsilon>0}A_{n,\epsilon}$...
0
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0answers
10 views

$\sigma$-algebra on space of measures

Let $P$ be the space of all probability measures on a Polish space $X$. Then, $P$ is Polish as well as if its topology is the weak*-topology (a.k.a. the topology of weak convergence). (EDIT: IS THAT ...
1
vote
1answer
105 views

Show that $\phi (\lambda^{d})=\lambda^{d}$

Let $d \in \mathbb N$ and define $\lambda^{d}$ as the lebesgue borel measure. Let $\phi: \mathbb R^{d} \to \mathbb R^{d}$ be measurable and $\phi(x)=-x$ Show that $\phi (\lambda^{d})=\lambda^{d}$ ...
0
votes
1answer
28 views

Does piecewise continuous imply Borel measurable?

It's extremely well-known that continuous functions are Borel measurable, but what about piecewise continuous functions? For the Lebesgue measure, I suspect that we'd have a proof as simple as "...
1
vote
1answer
50 views

Borel Bivariate Generating Function

I want to prove the following statement: $$ \beta(t,x)=C(1+t,x)= \frac {C((1+t)x)} {1-xC((1+t)x)} $$ Where $C(x)$ is the generating function for the Catalan Numbers and $ \beta(x) $ is the Borel ...
3
votes
1answer
36 views

uniformly continuous imply bounded

Proposition: Let $(X,d)$ be compact metric space, and $Y$ be Borel subset of $X$. Suppose $A$ is homeomorophic to $Y$. Then, uniformly continuous function $f:A \to \mathbb{R}$ is bounded function. I ...
1
vote
1answer
45 views

A question on Borel measurability

Let $X$ be a compact metric space. Given a map $x\mapsto E_x$ where $E_x\subset X$ is a Borel set in X. What can be said about the Borel measurability of the set $F_E:=\{(x,y)\in X\times X:y\in E_x\}$?...
2
votes
0answers
51 views

Why use transfinite induction?

Why use transfinite induction to prove? I think the inclusion relation is trivial by transfinite recursion. $\mathbf{11.B}$ The Borel Hierarchy Assume now that $X$ is metrizable, so that every ...
0
votes
1answer
26 views

Borel measurable function on the Euclidean space $\pmb{R}^4$

I want to know if my function $f$ is Borel measurable function or not. To be clear, I use the terms which are introduced in $\\$https://dspace.mit.edu/bitstream/handle/1721.1/14254/22712180-MIT.pdf?...
0
votes
0answers
31 views

$\sigma$-ideal of subsets of $2^\omega$

I do not understand why here on the page 148 $\cal M^*_{2, K}$ is a $\sigma-$ideal of subsets of $2^\omega$. I even do not know the weaker statement: why it is an ideal?
2
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2answers
24 views

A decreasing transfinite sequence of subsets of a countable set.

Let $X$ be a countable set and $(S_\alpha)_{\alpha< \rho}$ is a decreasing transfinite sequence of subsets of $X$ in the sense that $$ S_\alpha \supset S_\beta $$ whenever $\alpha<\beta$. Here $\...
1
vote
2answers
56 views

Prove Borel Sets

Proof $[a,b], (a,b], [a,b)$ are borel sets. I read from book, the definition of Borel sets : Borel sets is smallest $\sigma$-algebra that contains all open sets. But I cannot understand it. ...
0
votes
1answer
32 views

Equivalent definition of Borel $\sigma$-algebra on $\mathbb{R}$

I'm trying to show the following. Let $A\subseteq\mathbb{R}$. $\forall n\in\mathbb{N}\ (A\cap(n,n+1]\in\mathcal{B}((n,n+1])) \iff A\in\mathcal{B}(\mathbb{R})$, where $\mathcal{B}$ is the Borel $\...
0
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0answers
13 views

How to show that the next set belongs to Borel sigma algebra

Let $E$ a subset of $\rm I\!R$, and $a,b\in\rm I\!R$, and let $T(E)=aE+b=\{ax+b: x\in E\}$. Show that $T(E)\in\mathscr{B}(\rm I\!R)$ if and only if $E\in\mathscr{B}(\rm I\!R)$. I'm really lost with ...
0
votes
1answer
26 views

Prove the set for which a sequence of continuous functions converge is Borel

I've been trying to solve the following problem: Let $f_n\colon \mathbb{R} \to \mathbb{R}$ be a sequence of continuous functions. Prove that the set of points $E = \{x \in\mathbb{R}\mid\{f_n(x) \text{...
0
votes
2answers
72 views

Borel $σ$-algebra generated by intervals on real line

Show that Borel $σ$-algebra $B_{\mathbb R} = \sigma(A)$, when $A$ is: (a) $\{(a, b) : a, b \in \mathbb{R} \}$, (b) $\{(-\infty, a) : a \in \mathbb{R}\}$, (b) $\{(-\infty, a] : a \in \mathbb{R}\}$...
1
vote
1answer
43 views

Is the set of Borel measurable transformations the closure of continuous functions under pointwise limits?

If $S$ and $T$ are topological spaces, then $f:S\rightarrow T$ is called a Borel-measurable transformation if for every Borel set $B$ in $T$, $f^{-1}(B)$ is a Borel set in $S$. My question is, is the ...
1
vote
0answers
69 views

How to Modify a Borel function in a Borel way-Self study

I have the following questions: assume that $X$ be a standard Borel space (i.e. a Polish space equipped with the $\sigma$-algebra generated by open sets) with a (possibly Borel, invariant, ergodic) ...
0
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0answers
24 views

Is the projection of every closed set a Borel set?

Let $X$ and $Y$ be complete separable metric spaces and $A$ be a closed subset of $X\times Y$. Is it true that the projection of $A$ in $X$ is a Borel subset of $X$? Note that the projection of a ...
0
votes
1answer
28 views

How do I prove this for a borel sigma algebra on R?

I know that I much show each side is a subset of the other, but i am not sure where to start.
0
votes
1answer
59 views

Show that $f$ is $\mathbf{X}$-measurable iff $f^{-1}(E)\in\mathbf{X}$ for every Borel set $E$.

I'm trying to do exercises $2.N.$, $2.O.$, and $2.P.$ from Chapter $2$ of Bartle's The Elements of Integration and Lebesgue Measure.            &...
3
votes
0answers
126 views

Direct sum of subgroups of $\mathbb{Q}^n$ with $\mathbb{Q}$ is a Borel map - Self study

Let $\mathcal{Pow}(\mathbb{Q}^n)$ be the power set of $\mathbb{Q}^n$ and consider the product topology induced by the natural bijection $\mathcal{Pow}(\mathbb{Q}^n)\cong 2^{\mathbb{Q}^n}$ defined by $...
3
votes
2answers
129 views

How to check whether a set belongs to a $\sigma$-algebra? Not understanding “a Borel function”

Hello fine ladies and gentlemen, Now, I have never been very good with measure theory and I am struggling to be able to do the exercise. My definition of a sigma-algebra is the following. (...
1
vote
1answer
92 views

A consequence of the Selection Theorem for the Effros Borel space F(X) - self study

In Kechris' textbook "Classical Descriptive Set Theory", the following exercise is stated (pp. $77$, Exercise $(12.14)$): "Let X be a measurable space and Y a Polish space. Show that a function $f\...
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2answers
58 views

(T/F) If $\mu$ is a Borel measure on $\mathbb{R}$ and $A$ is a Borel set such that $\mu(A \cap K) = 0$ for all compact sets $K$, then $\mu(A) = 0$.

I am trying to prove or disprove the following statement: If $\mu$ is a Borel measure on $\mathbb{R}$ and $A$ is a Borel set such that $\mu(A \cap K) = 0$ for all compact sets $K$, then $\mu(A) = 0$. ...
0
votes
0answers
16 views

Haar measures are decomposable

The definition of decomposable measures is as follows: (Here $\mathcal{M}$ is a $\sigma$-algebra over $X$.) My question is part c) of the following exercise: I have managed to prove a) and b). For c)...
4
votes
1answer
79 views

Borel Hierarchy

i'm in trouble with an exercise on Kechris, Classical Descriptive Set Theory. The Theorem 22.4 shows $\Sigma_\xi^0(X)\neq\Pi_\xi^0(X)$ for each ordinal $\xi\lneq\omega_1$ and uncountable polish space $...
4
votes
1answer
46 views

Constructing sigma algebras in countably many steps

I'm learning about $\sigma$-algebras and was interested when my textbook briefly mentioned the impossibility of constructing the Borel $\sigma$-algebra of $\mathbb{R}$, $\mathcal{B}(\mathbb{R})$, from ...
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0answers
31 views

Showing that a multivariable function is Borel measurable. [closed]

I want to show that the following function is Borel measurable. Consider: $$f: \mathbb{R}^2 \rightarrow \mathbb{R}:\left\{ \begin{array}{ll} \sin\left(\frac{1}{x-y}\right) & x> y \\ ...
1
vote
2answers
93 views

$f$ measurable iff $f^2$ measurable and $\{f > 0\}$ measurable

I know for sure that if $f^2$ is measurable, that doesn't imply that $f$ is measurable, but how does the condition: $$ \{f > 0\} \text{ measurable }$$ play into making $f$ automatically ...
0
votes
1answer
35 views

reference request: Borel sigma-algebra of a subspace is the trace sigma-algebra

Let $\tau$ be the standard metric topology on $\mathbb{R}$. For $E\subseteq\mathbb{R}$, let $$\tau_E=E\cap\tau:=\{E\cap U:U\in\tau\}$$ denote the subspace topology on $E$. We write $\sigma(\tau)$ ...
1
vote
1answer
19 views

Finite-valued condition of measurable functions

In page 28 of Real Analysis by Stein, it is stated that if $f$ is finite-valued then it is measurable iff the sets $\{a<f<b\}$ are measurable for every $a,b \in \mathbb{R}$. I cannot understand ...
0
votes
1answer
52 views

Showing equivalence in a complex measure space

*I am trying to figure out how to answer this question in full. Recall, that if µ is a complex Borel measure on R, then the maximal function $M(µ) : \Re \to [0, 1]$ of µ is defined by $M(µ)(x) = \...