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.

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An example of an $L_p$ space with Borel measure and dimension larger than $\mathfrak c$

Given a normed space $X$, we can define the Borel $\sigma$-algebra in $X$ as the smaller $\sigma$-algebra containing all the open (or closed) sets of $X$. Suppose a measure $\mu$ is chosen in the ...
Emerick's user avatar
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3 votes
1 answer
226 views

Why is intersection of open sets not closed under countable unions?

I’m learning about Borel sets and I’m struggling to understand the following sentence. The intersection of every sequence of open subsets of $\mathbb{R}$ is a Borel set. However, the set of all such ...
lightweaver's user avatar
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23 views

Limit of the measure of the delta neighborhood of Borel Sets.

Statement 1.7 from Fractal Geometry by Falconer I am reading through Fractal Geometry Mathematical Foundations and Analysis by Falconer and I am not very familiar with measure theory. While reading ...
Pallav Pant's user avatar
0 votes
1 answer
42 views

Complement of a set of measure zero is dense in $[a,b]$?

Assume we have a subset of measure zero $N$ in the Borel-$\sigma$-field $\mathcal{B}([a,b])$. I think that $[a,b]\setminus N$ is a dense set. But how to prove that? My attempt: let $x \in [a,b]$ and ...
Perelman's user avatar
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Is the Borel $\sigma$-Algebra on a subset of $\mathbb{R}^d$ complete?

Why is the Borel $\sigma$-Algebra on a subset of $\mathbb{R}^d$ said to be incomplete wrt. to the Lebesgue measure? As I understand a complete measure space contains all subsets of all sets of measure ...
Perelman's user avatar
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0 answers
22 views

Complete, standard Borel spaces? (Complete Borel isomorphic probability spaces)

A common assumptions in probability theory is that the probability space is complete, i.e. if $\left(\Omega, \mathcal{A}, \mathbb{P}\right)$ is a probability space and $A \in \mathcal{A}$ satisfies $\...
user202542's user avatar
1 vote
0 answers
19 views

For a Borel subset $B$ of a complete, seperable metric space $S$ and $\epsilon > 0$, there exists compact $C \subset S$ with $P(B) < P(C) + \epsilon$.

For my bachelor thesis, I've been studying iterated random functions and a very limited amount of measure theory to understand it rigorously. One thing I could not understand is the following: Suppose ...
Steve's user avatar
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6 votes
1 answer
102 views

Borel and analytic sets - Why did Jech do this?

I've read in Jech's chapter on Borel and analytic sets that there is a universal $\Sigma_{\alpha}^0$ set $U \subseteq \mathcal{N} \times \mathcal{N}$ such that for every $\Sigma_{\alpha}^0$ set $A$ in ...
Link L's user avatar
  • 731
0 votes
0 answers
28 views

Why is transfinite recursion needed for the construction of Borel sets? [duplicate]

I'm reading the Hrbaceck and Jech's book and they say that recursion until $\omega$ does not work for constructing the Borel sets But I haven't been able to find such sequences, do you have any ...
Selena's user avatar
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Question About The Generation of The Borel $\sigma$-Algebra on $\mathbb{R}^d$

I am new to measure theory. I was asked to prove the following result: Proposition$\quad$ The $\sigma$-algebra $\mathcal{B}(\mathbb{R}^d)$ of Borel subsets of $\mathbb{R}^d$ is generated by each of ...
Grand Minister's user avatar
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Show that an affine subspace of $\mathbb{R}^d$ is a Borel set.

Everything is in the title. I was thinking of using the fact that an affine subspace of $\mathbb{R}^d$ is the set of solutions to a system of linear equations, but I can't figure out how to describe ...
Alex's user avatar
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5 votes
1 answer
139 views

Confused by this proof in Jech's set theory

In Jech's Set Theory, Chapter 11, the universal set $U$ is defined as: For each $\alpha \geq 1$, there exists a set $U \subset \mathcal{N}^2$ such that $U$ is $\Sigma_{\alpha}^0$ (in $\mathcal{N}^2$) ...
Link L's user avatar
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0 answers
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Lebesgue Measure of Closed and Open Sets

I am given the definition of the Lebesgue Measure: Lebesgue Measure $\mathcal P$ on the Borel $\sigma$-algebra $\mathcal B[0,1]$: $\mathcal P[[a,b]] = b - a$ for all $0 \leq a \leq b \leq 1$ The ...
asdf1234's user avatar
3 votes
0 answers
82 views

Extension of Borel map from a separable metric space to a Polish space

Suppose that $f:X\to Y$ is a Borel map from separable metric space $X$ to a $T_3$ space $Y$. Does there always exist a Polish space $\tilde X \supseteq X$ and $T_3$ space $\tilde Y\supseteq Y$ and an ...
Jakobian's user avatar
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1 vote
1 answer
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Separability of codomains of Borel functions taking values in completely regular spaces

I am looking for a reference (or a counterexample) to the following statement. Let $X$ be a separable metric space. Suppose that $Y$ is a completely regular topological space and $f\colon X\to Y$ is a ...
Tomasz Kania's user avatar
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6 votes
0 answers
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A maximal invariant generates the invariant $\sigma$-algebra

Let $G$ be a group action on the set $X$, and let $f:(X,\Sigma_X)\to (Y,\Sigma_Y)$ be a measurable maximal invariant ($f$ is constant on each orbit and take different values on different orbits). Am ...
Alphie's user avatar
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The cardinal number of a set from Cylindrical Sigma-algebra.

Let $\mathbb{R}^{[0,1]}$ be the set of all function on $[0,1]$, and $\mathcal{B}(\mathbb{R}^{[0,1]})$ be the sigma-algebra generated by all cylinder sets: $$\{ x=x(t):(x(t_1),\ldots,x(t_n))\in B \}$$ ...
eN.meshok's user avatar
1 vote
0 answers
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Borel sigma algebra of the Interval given by the restriction?

Let $\mathcal{B}$ denote the Borel sigma algebra. Is it true that $\mathcal{B}([0,1]) = \{A \cap [0,1] \ | \ A \in \mathcal{B}(\mathbb{R})$} ?
Mac Menders's user avatar
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Lebesgue measurement exercise together with Borel algebra

I am having difficulty proving that the lebesgue measure $\lambda$ in the question below is the only one that satisfies the productory property. I imagine I need to consider an arbitrary measure $\mu$ ...
Andre Luiz's user avatar
1 vote
1 answer
38 views

A non-Borel Lebesgue measurable set hat is a subset of a measurable zero Borel st

We know that the subset $N$ of $[0,1) \subset \mathbb R$ that contains exactly one member of the equivalence classes of $\sim$ defined on $[0,1)$, where $x \sim y$ iff $x - y$ is rational, is an ...
Squirrel-Power's user avatar
1 vote
0 answers
54 views

Measurability of $u(x-y):(\mathbb{R}^n\times \mathbb{R}^n, \sigma(\mathscr{L}^n\times\mathscr{L}^n)) \rightarrow (R,\mathscr{B}(\mathbb{R}))$

When we define the convolution on $L^1(\mathbb{R}^n)$, we are interested to proof that $\forall f,g \in L^1(\mathbb{R}^n)$ then $f\star g \in L^1(\mathbb{R}^n)$. In the proof of this we want use the ...
Manuel Bonanno's user avatar
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1 answer
29 views

Question about Schilling Exercise 4.21's solution

In the exercise 4.21(i) in the book Measures, Integrals, and Martingales, Let $X$ be a metric space and $\mu$ be a finite measure on the Borel sets $\mathcal{B} = \mathcal{B}(X)$ and denote the open ...
Thành Nguyễn's user avatar
5 votes
0 answers
57 views

Monotone (measurable) selection theorem

Given a Polish space $X$ and a set valued map $\Psi:X\to 2^X$ we way $\Psi$ is weakly Borel if for every open $U\subset X$, $$ \Psi^{-1}(U):=\{x\,|\,\Psi(x)\cap U\ne\varnothing\}\in\mathcal{B}(X). $$ ...
APP's user avatar
  • 76
1 vote
1 answer
47 views

Constructing measure using a non-monotonic function

It's well-known that we can construct a measure on $(\mathbb{R},\mathcal{B}(\mathbb{R}))$ using a Stieltjes measure function $F:\mathbb{R} \to \mathbb{R}$ which is a non-decreasing right continuous ...
S.H.W's user avatar
  • 4,379
0 votes
0 answers
75 views

Is the function Borel on $\mathbb{R}^2$?

Is the function $f(x, y) = \sum\limits_{n=1}^{\infty} \frac{[xn]}{1 + e^{n + [y]}}$ Borel on $\mathbb{R}^2$. I tried to estimate this function first, but I was only able to estimate it from above, and ...
Nick Schemov's user avatar
3 votes
1 answer
140 views

Do the normal numbers form a Borel set?

Normal numbers have a 'random' expansion. For example, in base 10 it means that all digits $0,1,\dots,9$ occur 'equally often' in its decimal expansion. A longstanding open problem is: is $\pi$ a ...
Riemann's user avatar
  • 783
0 votes
0 answers
69 views

All set in the Borel $\sigma$-algebra of $(\mathbb{R}, \tau_{usual})$, there are only $F_{\sigma}$ and $G_{\delta}$

Our teacher affirms this, but we have found a counterexample in the internet : $\mathbb{I}^{-} \cup \mathbb{Q}^{+} $ where: $\mathbb{I}^{-}=\mathbb{I} \cap (-\infty,0) $ and $\mathbb{Q}^{+}=\mathbb{Q} ...
James's user avatar
  • 1
-2 votes
1 answer
103 views

What is meant by " analytic set "?

I am a beginner with set theory. What is meant by " analytic set " ? I see this term in the context of set theory, calculus and real analysis but I have no idea what it means. See here : ...
mick's user avatar
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0 votes
0 answers
36 views

If $X \subset \mathbb{R}$ and $f:X \rightarrow \mathbb{R}$ the $X$is a Borel set because $X = f^{-1}(\mathbb{R})$

In Reference 1 (p. 33), Axler writes the statement: If $X \subset \mathbb{R}$ and there exists a Borel measurable function $f: X \rightarrow \mathbb{R}$, the $X$ must be Borel set [because $X = f^{-1}(...
Rene Girard's user avatar
0 votes
0 answers
49 views

Subset $V$ Lebesgue measurable, but not Borel measurable

Define the function $g: [0, 1] \rightarrow [0, 1]$ with $$g(y) := \inf\{x \in [0, 1] : f(x) = y\},$$ where $f$ is the Cantor function. Now let $V \subset [0, 1]$ be a set that is not Lebesgue ...
Minerva's user avatar
  • 153
2 votes
1 answer
69 views

$\sigma$-algebra generated by $\{ (a,b) : a,b \in \mathbb{Q} , a<b \}$

Is $\sigma$-algebra generated by $\{ (a,b) : a,b \in \mathbb{Q} , a<b \}$ the borel sigma algebra of $\mathbb{R}$. I wrote a proof and I am confused because I didn't find this result anywhere in ...
Ifielmodes's user avatar
0 votes
1 answer
49 views

Bounding the symmetric difference of a Borel set with a finite union of intervals

For $A \subset \mathbb{R}$ a bounded Borel set. Show that for all $\epsilon > 0$ there exists a set $U$ which is a finite union of intervals such that $\lambda_1(A \triangle U)\leq \epsilon$ With $...
Chocolatine's user avatar
0 votes
0 answers
48 views

Borel $\sigma$-algebra and $\sigma$-algebra generated by basis of topology

Let $(X, \tau)$ be a topology space, and assume the topology $\tau$ is generated by a basis $\beta$. Set $$ \mathcal{F} = \sigma(\tau) \equiv \sigma(\{O: O \in \tau\}) $$ and $$ \mathcal{G} =...
Thành Nguyễn's user avatar
0 votes
0 answers
39 views

Borel set under inner product is borel

Let $A \subset \mathbb{R}^d$ be a Borel set, $ e \in \mathbb{R}^d$ such that $||e|| = 1$ and define $P_e(a) = ea :\mathbb{R}^d \to \mathbb{R}$ the inner product. Show that $P_e(A)$ is Borel. We know ...
hteica's user avatar
  • 319
0 votes
1 answer
25 views

When does the Borel $\sigma$-algebra of compact convergence coincide with the product $\sigma$-algebra?

Let $X$, $Y$ be topological spaces, and $C(X,Y)$ the set of continuous functions $ X \to Y $, equipped with the compact-open topology. Let $\newcommand\Bco{\mathcal B_{\textrm{c-o}}} \Bco$ be the ...
Olius's user avatar
  • 514
0 votes
1 answer
43 views

Sigma ring generated by the class of compact subsets of X

In P Halmos measure theory book it written A Sigma ring S generated by C, the class of compact subsets of X where X is locally compact Hausdorff Space. Then S has open set and sets of S is called ...
Shubham Dhoria's user avatar
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0 answers
70 views

Definition of Borel regular outer measure.

I have ran across this in the hypothesis of one certain theorem. Let $X$ be a separable metric space with a Borel regular outer measure $\mu^{*}$ such that $\mu^{*}X = 1 $. I don't understand the ...
JohnNash's user avatar
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1 vote
0 answers
56 views

Can $\mathbb R$ be partitioned into two sets with positive measure on every interval? [duplicate]

An answer in this post got me wondering: Suppose $A \subseteq \mathbb R$ is Borel measurable, and let $B = \mathbb R \setminus A$. Is it possible that, for every nonempty open interval $I \subseteq \...
WillG's user avatar
  • 6,585
0 votes
0 answers
22 views

Questions regarding the proof for the set of real numbers that have a decimal expansion with the digit $5$ appearing infinitely often is a Borel set.

I have found two distinct proofs for the problem: Show that the set of real numbers that have a decimal expansion with the digit 5 appearing infinitely often is a Borel set. Both of which I could not ...
Dohyun Park's user avatar
2 votes
0 answers
25 views

For non second-countable TVS, is the sum of measurable functions again measurable? [duplicate]

Let $(\Omega, \mathcal A)$ be a measurable space and $E$ a topological vector space. Let $f,g:\Omega \to E$ be measurable. I already proved that Theorem $E$ is second-countable, then $f+g$ is ...
Akira's user avatar
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0 votes
0 answers
33 views

Extending a measurable function to Riemann sphere

For context I am looking at disjoint components of the Riemann sphere that map to one another injectively by a rational map $R$. i.e. we have the chain $$U_0 \xrightarrow{R} U_1 \xrightarrow{R} U_2 \...
OllyT777's user avatar
  • 125
2 votes
1 answer
85 views

The existence of measurable subsets in non-mesurable sets

Let $\mu$ be a Borel probability measure on $\mathbb{R}$ and let $S\subseteq\mathbb{R}$ be a Borel set such that $\mu\left(S\right)>0$. Is it true that for any $A_1,A_2\subset\mathbb{R}$ that are ...
user754663's user avatar
0 votes
1 answer
30 views

How to prove that $\{(s, t) \in \mathbb R^2 : s \le t \le s+1 \text{ and } t \in A\}$ is a Borel set?

I'm doing an exercise in which I need to apply Fubini–Tonelli theorem on a non-negative function $f$. So I have to check that $f$ is measurable. However, it turns out that applying the definition of ...
Akira's user avatar
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-1 votes
1 answer
79 views

Show that if $X,Y$ are topological spaces and $f:X\to Y$ is a continous function , then $f$ is a Borel measurable.

Show that if $X,Y$ are topological spaces and $f:X\to Y$ is a continous function , then $f$ is a Borel measurable. Any help what am I supposed to prove here ? My attempt: $\mathcal{B}(X) $ is the open ...
Algo's user avatar
  • 2,322
3 votes
1 answer
117 views

Exercise 1.7.15 from Salamon's Functional Analysis

This problem has three questions (a), (b) and (c). I've done most of them, with a little conclusion in (c) undone, which I've thought about it for a long time. (c) Let $f: X\to Y$ be a Borel ...
Matsuda's user avatar
  • 43
0 votes
1 answer
54 views

Countable generation of the Borel measure algebra

Consider the Borel sigma-algebra on $\mathbb{R}$ quotiented by the ideal of measure-zero sets (see definitions below). This forms a measure algebra. My question is whether this measure algebra is ...
psychicmachinist's user avatar
1 vote
0 answers
31 views

Whether the function $(x,x')\mapsto\rho\big(f(x),f(x')\big)$ is Borel for a Borel map $f\colon (X,d)\to (Y,\rho)$ with $(Y,\rho)$ not being separable?

Let $(X,d)$ be a compact metric space, let $(Y,\rho)$ be an arbitrary metric space. Let $\mathcal{B}(\mathbb{R}),\mathcal{B}(X),\mathcal{B}(Y),\mathcal{B}(X\times X), \mathcal{B}(Y\times Y)$ denote ...
Rafael's user avatar
  • 561
0 votes
1 answer
90 views

Intersection of a Borel set and its translation

My idea is to use regularity somehow and use compact sets where this statement clearly holds. This doesn't seem to be working. Is this the right approach? Any help is appreciated.
user avatar
2 votes
0 answers
29 views

Image of half-space through linear transform is Borel measurable

Let $T:(Z,\|\cdot\|_{Z})\to(H,\|\cdot\|_{H})$ be a continuous linear operator between a separable Banach space, $Z$, and a separable Hilbert space, $H$. Assume that both $Z$ and $H$ are infinite-...
Coco's user avatar
  • 161
0 votes
1 answer
66 views

Are these probabilities orthogonal?

Given $p\in (0,1)$ we define a Borel probability $\mu_p$ in the interval $[0,1]$. We assign $\mu_p([0,1/2))=1-p$ and $\mu_p([1/2,1))=p$. We iterate this process by diving each interval in two and ...
confusedTurtle's user avatar

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