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
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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
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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
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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
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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
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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
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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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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 ...
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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
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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
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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
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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
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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
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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
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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 ...
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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?
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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
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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
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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
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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
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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 ...
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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_{\...
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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
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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
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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 ...
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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}: ...
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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?
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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
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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.
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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
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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
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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
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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
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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
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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
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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 ...
