Stack Exchange Network

Stack Exchange network consists of 175 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers.

Visit Stack Exchange

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.

1
vote
0answers
40 views

How to decribe all bounded Borel functions where Lebesgue integral is zero

I'm working on an exercise and I'm not quite sure how to answer it. I need to describe all bounded borel measurable functions $f: [0,1] \to [-1,1]$ where $$ \int_{[a,b)} f d\lambda = 0 $$ There is a ...
1
vote
3answers
113 views

Finding Borel sets

Consider a function $f:\mathbb R\to\overline{\mathbb R}$ defined as $$f(x)=\begin{cases}\frac{1}{x}, x\neq 0\\ \infty, x=0 \end{cases}$$ Is $f$ Borel-measurable? I followed the answer given here ...
1
vote
1answer
50 views

Is the function Borel-Measurable?

Consider a function $f:\mathbb R\to\overline{\mathbb R}$ defined as $f(x)=\begin{cases}\frac{1}{x}, x\neq 0\\ \infty, x=0 \end{cases}$. Is borel-measurable? If we write $f$ as $f(x)=\frac{1}{x}\...
0
votes
0answers
17 views

Density of $C_c^0(X)$ with X LCH

I'm reading Folland Analysis textbook and to show that $C_{c}^{0}(X)$ is dense in $L^{p}(\mu)$ with $X$ a LCH (localy compact Hausdorff) and $\mu$ a Radon meausure. He said it is enough to ...
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 ...
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 $\pi$?
0
votes
1answer
28 views

Compactness of the set of finite Borel measures

Suppose $X$ is a compact subset of $\mathbb{R}^n$ for some $n \in \mathbb N$. Let $\mathcal M(X)$ denote the space of all finite Borel measures on $X$. Is $\mathcal M(X)$ compact under some commonly ...
0
votes
0answers
28 views

The measures which are non-negative on all convex non-negative functions

Let $X$ be a subset of $\mathbb{R}^n_{++}$ (vectors with non-negative coordiantes), and let $M$ be the set of regular Borel measures with finite variation on $X$ and finite first moment: $$M:=\{\mu;\ |...
0
votes
1answer
19 views

Why is $J_{n}(B):=\{j \in J: \mu(B \cap A_{j}) > \frac{1}{n} \}$ finite

Let $(X, \mathcal{A})$ be a measure space and $\mu : \mathcal{A} \to [0,\infty]$ a $\sigma-$finite measure. Let $J$ be a non-empty index set and $(A_{j})_{j\in J}$ a disjoint Family in $\mathcal{A}$ ...
0
votes
0answers
34 views

Proof about Borel measurable function

Let assume that the $X = [x_{1,1},x_{1,2}]\times[x_{2,1},x_{3,2}]\times[x_{3,1},x_{3,2}]\times[x_{4,1},x_{4,2}]$, $Y=[y_{1,1},y_{1,2}]$ where $x_{n,m} $ and $ y_{n,m}$ are real numbers for all $n,m$. ...
1
vote
1answer
15 views

Partitioning non atomic finite measure space

Let $(X,\Sigma, \mu)$ be a non-atomic, complete and finite measure space. I would like to know if the following is true: For every $\varepsilon \in (0, \mu(X))$ there are finitely many sets $X_1,...
0
votes
0answers
20 views

Factoring Variable in $L^2$

Suppose that $X,Y$ are random-variables in $L^2(\Omega,\mathcal{F},\mathbb{P})$, for some complete probability space $(\Omega,\mathcal{F},\mathbb{P})$, and suppose that there exists a Borel function $...
1
vote
1answer
36 views

A subset of a finite interval is Lebesgue measurable

Can you help me out to prove this? Let $[a,b]$ be a finite interval in $\mathbb{R}$ and $E \subset [a,b]$. Prove that if $m^*(E) + m^*([a,b]\setminus E) = b - a$, then $E$ is Lebesgue measurable. I ...
2
votes
0answers
23 views

Weak-continuity of an operator on the vector space of measures

Hello and thanks in advance for your time. Let $\mathcal{S} \subset \mathbb{R}^d$ (we can also choose $\mathcal{S}$ compact if this helps here) for some integer $d$ and let $M(\mathcal{S})$ be the ...
0
votes
1answer
40 views

Lebesgue-measurable or Borel-measurable

In practice, does one always use Lebesgue-measurability? So do we always use Lebesgue-measurable instead of Borel-measurable on $\mathbb{R}^n$ because it is usually more convenient?
0
votes
1answer
93 views

Difference between Lebesgue Sigma Algebra and Borel Sigma Algebra

I have read that probability measure cannot be defined on set of all subsets of unit interval namely (0,1]. Proof uses construction of the Vitali set etc. Specifically, it is well known that ...
1
vote
2answers
51 views

Lebesgue-measurable almost everywhere equals Borel measurable function [closed]

Let $\, f: \mathbb{R}^n \rightarrow [-\infty, +\infty]$ be a Lebesgue measurable function. I want to show that f is $\lambda_n$- alomost everywhere equal to to a Borel measurable function $\, f'$. ...
1
vote
2answers
97 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 ...
4
votes
1answer
80 views

Regularity of measure in Lemma 7.2.6 of Bogachev

In the book "Measure Theory" of Bogachev, vol. 2, Lemma 7.2.6 states the following. Let $\mu$ be a $\tau$-additive, regular Borel measure on a topological space $X$, and let $\{f_\alpha\}$ be an ...
2
votes
1answer
37 views

function continuous ae but not borel measurable

It is easy to prove that if $f$ is a function continuous almost everywhere, then $f$ is Lebesgue-measurable by using the property that $\mathcal L$ (the Lebesgue-measure) is complete. Though I've ...
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) = \...
0
votes
1answer
57 views

Borel Measurable Set Related to Sections

Let $E\subseteq\mathbb{R}^{n+1}$ be a Borel set and define \begin{equation} E^x=\{y\in\mathbb{R}\colon(x,y)\in\mathbb{R}^{n+1}\} \end{equation} for $x\in\mathbb{R}^n$, where we identify $\mathbb{R}^{n+...
2
votes
0answers
27 views

Outer approximation of Borel sets by disjoint union of sets of the form $(\frac{k}{n}, \frac{k+1}{n} ]$

Original question: $f$ is a Borel measurable function on $(0,1]$ such that $\int_{(\frac{k}{n},\frac{k+1}{n}]} f = 0$ for all $n\in \mathbb{N}$ and $0 \le k \le n-1 $ . Then to show that $f$ is zero ...
0
votes
0answers
14 views

Finding a Borel-measurable inverse

Let $E$ and $F$ be Borel subsets of $\mathbb{R}$. We say that $m:E\to F$ is measure-preserving (with respect to the Borel measure on $\mathbb{R}$) whenever $B\subseteq F$ is Borel in $\mathbb{R}$ ...
0
votes
1answer
22 views

A double integral being finite implies the corresponding Borel-measure is $= 0$ on singletons

Let $\mu$ be a positive Borel measure on $\mathbb R^d$ with $0 < \mu(\mathbb R^d) < \infty$. Let $I_s(\mu) < \infty$, where $I_s(\mu)$ is defined by $$I_s(\mu) := \int_{\mathbb R^d} \int_{\...
1
vote
2answers
88 views

A counterexample to the epsilon-delta criterion for Absolute Continuity of Measures

Let $p>0$, and let $\mu$ be a Borel measure on $[0,\infty)$ defined by $\mu(E)=\int_Ex^pd\lambda$ where $\lambda$ denotes Lebesgue measure. Show that $\mu$ is absolutely continuous with respect to ...
1
vote
2answers
58 views

$f, 1/f$ integrable implies that $f^2$, $1/f^2$ integrable?

on the measure space $(X,A,m)$ let $f: X \to \mathbb R$ be Borel-measurable, it is given that $f$, $1/f$ are well-defined and integrable. I would like to prove that this implies that $f^2$ and $1/f^...
1
vote
0answers
50 views

Lusin's theorem on arbitrary measure space

This question is generalization of Lusin's theorem, though it appears long. Let $(X, \rho)$ and $(Y, \sigma)$ be metric spaces with $(Y, \sigma)$ separable, and let $\mu$ be a finite Borel measure ...
0
votes
1answer
113 views

How to define the Lebesgue measure on the real line

I am confused by notes we previously made. Let $\mathcal{B}^{d}$ be the Borel sigma-algebra and $\lambda^{d}$ be the Lebesgue-Borel measure. It was defined as follows: For all $Q:=\times_{i=1}^{d}[a_{...
2
votes
1answer
40 views

Regular Measures

Let $\Sigma$ be the Borel $\sigma$-algebra of $\mathbb{R}$. We define a measure $\mu$ on $\Sigma$. For every Borel set $A$ we define $\mu(A)$ to be the number of elements from the set $\{\frac{1}{n}: ...
-1
votes
1answer
44 views

Is the function $f(x) = \lfloor x\rfloor$ is measurable? [closed]

Let $(R, B)$, where $B$ is the Borel $\sigma$-algebra, be our measurable space. In this measure space how do you that the function $$f(x) = \lfloor x\rfloor$$ is measurable?
0
votes
1answer
14 views

Tips to show that the Random Variable $X$ is $\mathcal{B}(\mathbb R)-\mathcal{B}(\mathbb R)-$measurable

Let $X: \mathbb R \to \mathbb R$, where $\forall \omega \in \mathbb R -\mathbb Q: X(\omega)=0$. Show $X$ is $\mathcal{B}(\mathbb R)-\mathcal{B}(\mathbb R)-$measurable function: My ideas: Using the ...
0
votes
2answers
32 views

Show $\mathcal{B}( \mathbb R)$ contains all countable subsets of $\mathbb R$

We defined $\mathcal{B}(\mathbb R):=\sigma(\{[a,b[\}),$ where $a < b \in \mathbb R.$ I have been able to prove: i) for any $b \in \mathbb R$, $\{b\} \in\mathcal{B}(\mathbb R)$ and subsequently ...
0
votes
0answers
79 views

Lebesgue Measure equal to any translation invariant Measure on the Borel sigma Algebra

I have come across a question in my measure theory textbook and can pretty much figure out how to construct an answer to the second part but not the first part. Any solutions or hints would be ...
0
votes
1answer
56 views

When sigma algebra of product topology equals sigma algebra of projections

Given a locally compact Hausdorff space $X$, let $S:= X^\mathbb{N}$ the corresponding product topology. I am trying to prove the following claim : $\sigma(S) = \sigma(p_i)$ where $\sigma(S)$ is the ...
0
votes
1answer
34 views

A measure for which Lusin's theorem fails?

Lusin's theorem states that if $f:[a,b]\rightarrow\mathbb{R}$ is a Lebesgue measurable function, then for any $\epsilon>0$ there exists a compact subset $E$ of $[a,b]$ whose complement has Lebesgue ...
1
vote
1answer
23 views

Does there exists a set which has the exterior measure,but doesn't Lebesgue measurable?

Does there exists a set which has the exterior measure,but doesn't Lebesgue measurable? Can one give a example of this ? I do really puzzled.
1
vote
0answers
42 views

Finding a sequence of Borel measurable functions whose limit exist but Lebesgue integrals differ

I want to find a sequence of functions say $f_1 , f_2, \ldots$ such that each $f_n : [0,1] \to [0,+\infty)$ is a Borel measurable function with the limit $f(x)=\lim_{n \to \infty} f_n(x)$ existing for ...
0
votes
1answer
45 views

Proof of factorization lemma

Let $X$ be a set, $(Y, \mathbb A)$ a measurable space and $g: X \to Y$ a surjective function. We consider on $X$ the $\sigma$-algebra induced by the function $g$ ($g^{-1}(\mathbb A))$. If the function ...
0
votes
0answers
30 views

Existence of unique measure

Let $F: \mathbb R \to \mathbb R$ be a continuous, bounded, non-decreasing and such that $\lim_{x \to - \infty} F(x) = 0$. Can anybody show me why there exists a unique (!) measure $m_F$ on $(\mathbb ...
1
vote
0answers
31 views

Proving a Borel Measure on the unit circle.

My professor asked this question: Let $E \subseteq \mathbb{S}^{n-1}$ be a Borel set and let $E_1 = \{x\in \mathbb{R}^n \setminus \{0\}:|x|\in (0,1], \frac{x}{|x|} \in E\}.$ Show that $\sigma(E) = nm(...
2
votes
1answer
59 views

A continuous Function from $R$ to a Banach Space is Borel-Measurable

I think the notation of my book is a bit odd, so I'm having trouble finding any other sources to help me with this proof. The thing I'm trying to prove is that a continuous function, $f$, from $R$ to ...
0
votes
1answer
116 views

Sequence of Borel measurable functions where the limit of the integral of the sequence is not equal to the integral of the limit of the sequence

Hello I am trying to think of two different examples of sequences of Borel Measurable functions $f_n(\omega)$ where $\lim_{n\to\infty} \int\limits_{\Omega} f_n du > \int\limits_{\Omega} (\lim_{n\...
1
vote
0answers
49 views

What is a good intuition of a Borel-measurable set?

Is there a good intuition/explanation of a Borel-measurable set in terms of probability theory?
1
vote
1answer
85 views

Is the Borel sigma algebra complete under some measure?

A measure space $(X,F,\mu)$ is complete if whenever $\mu(E)=0$, every subset of $E$ is an element of $F$. Every incomplete measure space has a completion, which extends the sigma algebra to more sets ...
1
vote
1answer
35 views

Measures and Borel Sigma algebras

Let $X$ be a separable metric space and $B(X)$ the Borel Sigma algebra. In addition, we have $m$ as a measure on the measurable space $(X, B(X))$. If $Z = \{w$ open : $m(w)=0\}$. Why is Z NOT empty (...
0
votes
1answer
68 views

Inequivalent definitions of the support of a Borel measure

All definitions appearing in this question apply to Borel measures. Many sources (including Wikipedia and Folland) define the support of a Borel measure to be the set of points whose open ...
0
votes
0answers
73 views

Example of a Borel measurable function

I am reading Borel measurable function in the analysis independently, and I stuck on a problem. Hope you guys will help me out. Give an example (with explanation) of a Borel measurable function $f:[0,...
0
votes
1answer
25 views

Determining whether an interval (0,1] $\in\mathfrak B(\mathbb R)$ or not.

Determine, with justification, if the interval $(0,1]$ $\in\mathfrak B(\mathbb R)$ or not, given $\mathfrak B(\mathbb R)$ is the Borel $\sigma$-algebra on $\mathbb R$, which contains all the the open ...