Questions tagged [borel-measures]

Use this tag for questions related to Borel measures, which, on a topological space, are measures defined on all open sets.

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31 views

Convergence of total variation given uniform convergence of variation in open sets

Let $\Theta$ be a metric space, and let $\{\mu_\theta\}_{\theta \in \Theta}$ be a collection of probability measures over the Borel $\sigma$-algebra $\mathcal B(X)$ of some metric space $(X, \rho)$. ...
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0answers
38 views

Is there a converse to Caratheodory's criterion?

Caratheodory's criterion states the following: Let $\mu$ be an outer measure on $\mathbb{R}^n$. If for all sets $A,B\subseteq \mathbb{R}^n$, we have $\mu (A\cup B) = \mu(A) + \mu(B)$ whenever $\text{...
2
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1answer
30 views

$\mathcal{B}(\mathbb{R})$-measurable vs. $\mathcal{B}(\overline{\mathbb{R}})$-measurable

Let $(E,\mathcal{E})$ a measurable space and $f \colon E \to \mathbb{R}$ a function. Is it right, that $f$ is $\mathcal{E}$-$\mathcal{B}(\mathbb{R})$-measurable iff $i \circ f$ is $\mathcal{E}$-$\...
3
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1answer
22 views

Is a Radon measure always positive on non-empty open sets?

A measure is locally finite if it is finite on all compact sets from the underlying $\sigma$-algebra. A measure is regular if every measurable set $A\in\Sigma$ can be approximated from above by open ...
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1answer
15 views

Prove continuity of a function constructed from a Borel measure

If $\mu$ is a finite Borel measure on $\mathbb{R}$, then the function $F^{\mu}: \mathbb{R} \rightarrow \mathbb{R}$ , $F^{\mu}(\lambda) = \mu((-\infty, \lambda])$ is right-continuous. Attempt at ...
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1answer
19 views

Is the set $J^n := \{ (x,y)\in \mathbb{R}^2: |x| > m_n, y = h(x)\}$ measurable?

Im currently working on a proof regarding APARCH-models, where the unction $h$ in the following context comes from: Given is the set $J^n := \{ (x,y)\in \mathbb{R}^2: |x| > m_n, y = h(x)\}$, where $...
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0answers
19 views

Prove Borel measurable function

I want to prove that $\mathcal{f:[0,4\pi] \to \mathbb{R}:f(x)} = \left\{ \begin{matrix}\mathcal{\frac{1}{1+\sin(x)}} & \sin(x) \neq-1\\ 0& \sin(x)=-1 \end{matrix}\right.$ is a Borel measurable ...
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1answer
32 views

Is the unit circle in $\mathcal{B}(\mathbb{R}^2)$

Is the unit circle in $\mathcal{B}(\mathbb{R}^2)$ ? The definitions I am working with is that $\mathcal{B}(\mathbb{R}^2)$ is the $\sigma$-Algebra generated by open (equivalently closed, or clopen) ...
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1answer
37 views

Does convergence in measure imply “supports getting close” (made precise in question body)?

In connection to my question here, let $P_{n}(n=1,2, \ldots)$ on $([0,1], \mathcal{B}([0,1]))$, where $\mathcal{B}([0,1])$ is the Borel $\sigma$-algebra on $[0,1]$, be a sequence of measures which ...
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1answer
67 views

How to show that $\lambda(D)=\infty$?

I have a bit trouble showing that $\lambda(D)=\infty$. I have been giving the following problem: Given $D=\{(x,y)\in \mathbb{R}^2 | 0<x=y<1\}$ and $\mathcal{N}=\{N \in \mathcal{B} | m_I(N)=0\}$ (...
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1answer
16 views

Absolutely continuous measures

We have two measures with $\mu_F(]-\infty,x])=F(x)$ $\mu_G(]-\infty,x])=G(x)$ $G(x)=F(x)^2$ Prove that $\mu_G$ is absolutely continuous with respect to $\mu_F$. So we have to prove that $\mu_F(A)=0\...
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2answers
70 views

transformation for my self

I am trying to prove this equation of integral transformation with Borel measure. But so far have no idea. Can you help me how to start. Suppose $(X, A, \mu)$ is a $\sigma$-finite measure space and $f:...
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1answer
25 views

Show that an integral function is a Borel function

I've been trying to prove that the Hardy-Littlewood Maximal function is a Borel Function for a while now (and was hoping to get some help) and have been stuck on the final step for a while, which is ...
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1answer
30 views

Problem about almost everywhere convergence in measure theory

I am having trouble with the following problem Let $(X, \mathcal{F}, \mu)$ a measure space where $\mu (X)<\infty.$ Let $f,f_n:X \to \mathbb{C}$ be measurable. Set $A_n=\{ |f_n-f|\geq a_n\}$ where ...
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1answer
39 views

$\sigma$-field generated by Borel sets

I what sense in the snippet is the $\sigma$-field $\cal M$ on $M$ generated by all the functions $$m\to m(C)$$ for $C$ Borel;how does $\cal M$ look like ?
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23 views

Is every Lebesgue-Stieltjes measure $\sigma-$finite?

Let $F:\mathbb{R}\longrightarrow \mathbb{R}$ be increasing and right continuous. Then one can define $\mu_F$ the Lebesgue-Stieltjes measure induced by $F$, by setting $μ_F(a,b]=F(b)−F(a)$ for half-...
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0answers
15 views

Measures with common distribution function

Let $\varphi: [0, \infty) → [0, \infty]$ be growing monotonously. $\tau_1$ and $\tau_2$ are two (finite) measures on $([0, \infty), B([0, \infty)))$ and $\varphi$ is their distribution function. how ...
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1answer
27 views

Find borel measure for on $X=[1,\infty)$ such that for every $s\geq 1$ the following holds $\mu(T_sA)=\mu(A)$ for every measureable $A$.

For every $s\geq1$ we define the transformation $T_s(x)=s\cdot x$. Find a borel measure which is not the zero measure on $X=[1,\infty)$ such that for every $s\geq 1$ the following holds $\mu(T_sA)=\mu(...
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0answers
22 views

Borel measure that is locally finite but not borel regular

I was wondering if there is an obvious example of a borel measure $\mu$ on $\mathbb{R}^{n}$ that is locally finite but not borel regular, i.e. if there exists $E\subset\mathbb{R}^{n}$, such that for ...
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0answers
20 views

How to prove that the dual of $L^\infty$ is the set of bounded finitely additive measures?

How to prove that the dual of $L^\infty(X,\mu)$ is the space of finitely additive, signed measures on $X$ that are absolutely continuous with respect to $\mu$? I only find that it's a "well known ...
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1answer
43 views

Does the set of open sets in $\mathbb{R}$ is countable?

I want to prove that the cardinality of Borel $\sigma $ algebra is $\aleph$, using the next proposition: If $ E \subset P(X) $ is infinite, and the cardinality of E is $\aleph $ , the $\sigma$ algebra ...
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1answer
28 views

If $f(x)$ is a Borel function, is $f(x+a)$ also a Borel function?

I've been recently covering Borel functions and wondering are they preserved by the following property. If $f(x)$ is a Borel function then $f(x+a)$ is also a Borel function where $a$ is a real number. ...
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1answer
33 views

Show that monotone function f: I->R is measuable on an interval

Show that a monotone function f: I -> R is borel-measureable. I know, that this question has been asked and answered multiple times here, but only for f: R->R und not for f: I->R. The first ...
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1answer
48 views

Prove the following inequality for non-negative Borel measurable functions defined on $\Bbb R.$

Let $f : \Bbb R \longrightarrow [0,\infty)$ be a Borel measurable function. Show that $$\displaystyle{\sum\limits_{n = 1}^{\infty} \text {m} \left (\left \{f \gt n \right \} \right ) \leq \int f\ \...
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0answers
43 views

Getting Borel-measurability from a product measure space

I have that the graph of a function $G(f) = \{ (x, f(x)) : x \in \mathbb{R} \} $ for $f : \mathbb{R} → \mathbb{R} $ is such that $G(f) ∈ B \otimes B$. Here $B$ is the $\sigma$-algebra of Borel sets. ...
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1answer
22 views

Equivalent measures and their topological supports

The topological support $\sigma(\mu)$ of a Borel measure $\mu$ can be defined as the smallest closed Borel set $X$ that supports $\mu$, in the sense that $\mu(\mathbb{R}\setminus X)=0$. One can show ...
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1answer
53 views

Two sigma-algebras coincide

We have a polish space $\mathbb{R}^{+\infty}$ with a metric $\rho(x, y) = \sum_{k=1}^{+\infty} 2^{-k} {|x_k - y_k| \over 1 + |x_k - y_k|}$. Show that in this space Borel $\sigma$-algebra coincides ...
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2answers
43 views

Are slices of Borel sets also Borel sets?

Suppose E is Borel in $\mathbb{R}^{a+b}$. Show that the slice $E^{x_1}$={$x_2\in\mathbb{R}^b|(x_1,x_2)\in E$} is Borel for each $x_1\in \mathbb{R}^a$. What I have so far: First, any sigma algebra ...
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1answer
31 views

Showing $\mu_f$ is a measure on the $\sigma-$algebra $\mathcal{B}$ of Borel subsets of $\mathbb{R}.$

$\def\R{{\mathbb R}}$ Can I please receive help proving this? Thank you. Suppose $(X,\mathcal{M}, \mu)$ is a complete measure space and $f: X\to \R$ is a measurable function. Let $\mu_f(B) = \mu(f^{-1}...
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1answer
37 views

Is $f$ a Borel measurable function?

I have define $f$ by $f=\frac{1}{b}$ if $x \in \mathbb{Q} \cap [0,1]$ and $x=\frac{a}{b}$ in lowest terms. $f=34$ if $x \in [0,1] \backslash \mathbb{Q}$. Is $f$ Borel measurable? I know I need to show ...
4
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2answers
60 views

Prove that $f$ is one-to-one on $B$ and $f^{-1}:Y\to B$ is Borel.

Let $f: X\to Y$ be a continuous and surjective map between compact metric spaces. Prove that there is a Borel set $B\subset X$ such that $f(B)=Y$, $f$ is one-to-one on $B$ and $f^{-1}:Y\to B$ is Borel....
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0answers
40 views

Nontrivial Signed Measure on Lebesgue Measurable Sets Being Trivial on Borel Sets

Let $\mathfrak{L}(\mathbb{R})$ be the collection of Lebesgue measurable sets and $\mathfrak{B}(\mathbb{R})$ be the Borel sets. Question: Is there a nontrivial signed measure on $\mathfrak{L}(\mathbb{R}...
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1answer
52 views

A problem about measure theory, sigma algebra and Borel sigma algebra

Problem: Let $\mathcal{S}=\{(-b,b): b\geq 0\}$. Is $\sigma(\mathcal{S})=\mathfrak{B}o(\mathbb{R})$? Notation: $\sigma(\mathcal{S})$ is the sigma algebra generated by $\mathcal{S}$ and $\mathfrak{B}o(\...
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0answers
23 views

Definitional Question: Can a Borel Measure be Defined on More Than the Borel Sets?

On Wikipedia, a Borel measure is defined as any measure that is defined on the Borel $\sigma$-algebra of a locally compact Hausdorff topological space. Obviously if you have a measure $\mu$ that is ...
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1answer
45 views

prove that A does not generate the $\sigma$-algebra on $\mathbb{R}$

Let's say $A=\{[-a,2a]\mid a\in[0,\infty)\}$. I want to prove that it doesn't generate the $\sigma$-algebra on $\mathbb{R}$. I have no idea how to start this prove because I thought it would produce ...
0
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1answer
42 views

If both $f \ge 1$ and $f\le 1 $ almost surely then $f = 1_A$, where $A$ is borel

Let $f:[0,1]\rightarrow \mathbb{R}$ be a positive measurable function. If $f\leq 1$ almost surely and $f\geq 1$ almost surely (with respect to lebesgue measure). Show that $f=1_{A}$ for some $A\in B[0,...
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0answers
9 views

Positive definiteness wrt different borel measures

Let $K$ be a compact set in $\mathbb R^d$. Let $dx$ represent the usual Lebesgue measure and let $Q$ be a compact strictly positive definite integral operator $L^2(K,dx)\to L^2(K,dx)$: $$Q\phi(x) = \...
2
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2answers
90 views

Show that two measures are the same

I have that $u$ and $v$ are two measures on $(\mathbb{R},\mathscr{B}(\mathbb{R}))$ and $u=((-\infty,a]) = v((-\infty,a]) < \infty $ for all $a\in \mathbb{R}$ and I want to show that $u=v$. I know ...
2
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1answer
93 views

Show that two Lebesgue measures are the same

I have the following sets: $$ \begin{align} A = \{(x,y)\in \mathbb{R}^{2}|x\geq0, y\geq 0, |x|^{3}+|y|^{3} < 1 \}, \\ B = \{(x,y)\in \mathbb{R}^{2}|x\leq0, y\geq 0, |x|^{3}+|y|^{3} < 1 \}...
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1answer
75 views

How to find intervals of a Borelfunction

I have that $u:\mathbb{R} \rightarrow \mathbb{R}$ given as $$ u(t) = \begin{cases} 3 & t<0\\ t & t\geq 0 \end{cases} $$ and I want to find $\{u\geq a\}$ for all $a \in \...
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0answers
76 views

Restriction of regular measures

I'm studying Measure theory at university. For practising with some exercises, I found some available notes on the web. While reading chapter about Borel measures, the following definitions are given: ...
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1answer
226 views

Showing if $f$ is Borel measurable and $B$ is a Borel set, then $f^{-1}(B)$ is a Borel set.

The following problem is from Royden & Fitzpatrick (4 ed.). I am stuck on showing (ii), can someone please help me prove it? Thank you. $\def\R{{\mathbb R}}$ Page 59, problem 8. (Borel ...
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0answers
46 views

Do finitely additive measures solve the “problem of measure”?

Does there exist a unique function $\mu$ satisfying the following properties? $\mu:\mathcal P(\mathbb R)\to [0,\infty]$ $\mu(A+x)=\mu(A) \qquad\qquad$ for all $A\in\mathcal P(\mathbb R),x\in\mathbb R$...
2
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1answer
48 views

borel measure on a set

Let $f: \mathbb{R}^n \to \mathbb{R}$ be right-continuous ($f(x) = \lim\limits_{h\to 0^+} f(x+h) \,\forall x \in \mathbb{R}^n$) and $n$-increasing (i.e. $\Delta_{(a,b]} F \geq 0\,\forall a \leq b$ (...
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1answer
28 views

Prove that the support of $\mu$ is compact.

Let $\mu$ be a measure on the Borel $\sigma$-field of $\Bbb R$ such that $\mu (\Bbb R) = 1.$ Recall that the support of $\mu$ is the largest closed set $C$ such that for all open sets $U$ with $U \cap ...
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0answers
22 views

Understanding the definition of convolution in Bauer's book

Am trying to understand the the definition of convolution in Bauer's measure theory book. He writes: Let $\mathcal{M}^d$ denote the set of all finite Borel measures on $\mathcal{B}^d$. Initially we ...
2
votes
1answer
42 views

Is it possible to approximate any Borel measure with Dirac's Deltas?

Given a complete separable metric space $(X,d)$. My question is if it is true that for any finite positive Borel measure $\mu\in\mathcal{M}^+_b(X)$ there exists a sequence of finite atomic measures $...
2
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1answer
112 views

Show that $\nu(E) = \int_E \phi \,d \mu$ is inner and outer regular.

Let $\phi \geq 0$ be a function in $L^1(\mu)$ where $\mu$ is a Radon measure (= a Borel measure on $X$ that is finite on compact sets, inner regular on open sets and outer regular on all compact sets) ...
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1answer
32 views

Folland exercise: measure on the countable ordinals

I managed to prove $(a),(b),(c),(d),(e)$ from the following exercise in Folland's book: I'm currently attempting $(f)$. In particular, I am stuck at showing that $\mu$ is countably additive. So let $(...
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0answers
26 views

Example of borel measurable function for which there doesnot exist a borel set such that function restricted on it is continuous

Give an example of a Borel measurable function f from R to R such that there does not exist a set B ⊂ R such that $| R \setminus B |$ = 0 and $f|_B$ is a continuous function on B. By Lusin's theorem ...

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