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.
472
questions
-5
votes
0
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39
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Any set of $R^d$ is $G/delta$ or $F/sigma$ [closed]
is this true? We have to try it and we don't know where to start.
2
votes
1
answer
51
views
Are finitely additive Baire measures the same as Radon measures
If $X$ is a compact Hausdorff space, we know that finite Baire measures uniquely extend to a Radon measure. Furthermore every Radon measure also restricts to a Baire measure. We also know that every ...
1
vote
0
answers
23
views
$\mu_0$ (semi-finite) is outer regular on all borel with finite measure
Let $\mu$ be a Radon Measure on $X$ and let $\mu_0$ be the semifinite part of $\mu$ ($\mu_0(K) = \sup \{\mu(F), F \subset K , \mu(F) < \infty \}) $
a) Show that $\mu_0$ is inner regular in all ...
0
votes
0
answers
48
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 ...
0
votes
0
answers
25
views
Equivalence of Assertions for Functions of Bounded Variation
I am trying to prove that the following three assertions about a function $f$ on an interval $[a, b]$ are equivalent, but I am encountering some difficulties:
$f$ is the difference of two increasing ...
0
votes
0
answers
64
views
Question on application of Borel-Cantelli lemma
I have an exercise in my measure theory class which goes as following:
If $(X,A,\mu)$, where $A$ is a $\sigma$-algebra, is a measure space and $(B_n),n=1,2,\ldots$ a sequence of sets in $A$. Show that ...
0
votes
1
answer
40
views
Show that the counting measure $\zeta$ on $\mathbb{R}^{N}$ is Borel regular. [closed]
I want to show that the counting measure $\zeta$ on $\mathbb{R}^{N}$ is Borel regular.
The counting measure is defined as follows:
In addition, I know that the counting measure $\zeta$ is not $\sigma$...
2
votes
0
answers
58
views
For sets $B$ with $μ(B)<∞$ the existence of an $μ$-hull follows from the Borel regularity of the measure $μ$
From this I should also be able to conclude that for sets $B$ with $\mu(B)<\infty$ the existence of an $\mu$-hull follows from the Borel regularity of the measure $\mu $:
I know:
An outer measure $...
0
votes
0
answers
21
views
Pushforward measure is Borel regular
Let $\mu$ be an outer measure in $\mathbb R^n$ and $f:\mathbb R^n \to \mathbb R^m$ a function i know that if $\mu$ is Radon and $f$ continuous and proper then the pushforward measure $f_{\sharp}(\mu)$ ...
0
votes
1
answer
56
views
Which subsets of $ \mathbb{R}^{N} $ are measurable with respect to the dirac measure?
Let $ a \in \mathbb{R}^{N} $ and $ \delta_{a} $ be the corresponding
Dirac measure. Which subsets of $ \mathbb{R}^{N} $ are measurable with
respect to $ \delta_{a} $? Is $ \delta_{a}$ Borel-measure,
...
0
votes
1
answer
15
views
Composition of Borel measurable and Labegsue mesurable
If $f$ is finite and Lebesgue measurable on $R^n$ and $\phi$ is Borel measurable on $R^1$, then $\phi∘f$ is Lebesgue measurable.
For any open set $G,$ we have that $\phi^{-1}(G)$ is borel measurable. ...
0
votes
0
answers
36
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 ...
1
vote
0
answers
21
views
Is the canonical form of a measurable function measurable?
Let $(E,\mathcal{E})$ be a measurable space, and let a positive function $f:E \rightarrow \mathbb{R}$ be defined as
\begin{align}
f=\sum_{n=1}^{\infty}a_n\mathbb{I}_{A_n}
\end{align}
with $(a_n)_{n=1}^...
0
votes
2
answers
71
views
Equality of Integrals implies equality of measures
I have some trouble solving the following problem:
Let $\mu, \nu$ be two measures on the Borel-$\sigma$-algebra on $\mathbb{R}$. Assume that $\mu(\mathbb{R}) = 1$. Prove that, if for all continuous, ...
7
votes
1
answer
88
views
Why inner regularity of measure is defined as to be approximable from within by compact sets and not by closed sets?
This question has been asked here, but I don't believe that we have had a satisfactory answer, so I would like to reformulate that question. Please forgive me if this question turns out to be rather ...
1
vote
0
answers
25
views
Terminology and notation for the restriction of the Lebesgue measure on $\mathbb{R}^n$ to the Borel subsets of $\mathbb{R}^n$
Let $m^n: \mathcal{L}^n \to [0,\infty]$ be the Lebesgue measure on $\mathbb{R}^n$, and let $\mathcal{B}^n$ denote the collection of Borel subsets of $\mathbb{R}^n$. Is there a name for the measure $m^...
2
votes
0
answers
89
views
Measure-preserving transformation from separable metric space into an interval $[0;1]$.
Let $X$ be a separable metric space with a Borel regular outer measure $\mu^{*}$ such that $\mu^{*}X = 1 $.
I want to prove that there exists a measure preserving transformation $f : X \to [0;1]$ ...
0
votes
1
answer
27
views
Support of a measure on Compact Hausdorff space
I've find this statement in Wikipedia page about support of measures:
Let $(X,\mathcal{B})$ a compact Hausdorff topological space equipped with Borel $\sigma$-algebra. If $\mu$ is a measure on $\...
0
votes
0
answers
65
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 ...
3
votes
1
answer
90
views
Empirical distribution learns w.r.t total variation distance
I am trying to prove or disprove that the empirical distribution can learn any continuous distribution w.r.t the total variation distance. The context is the one of statistical learning.
I am quite ...
2
votes
1
answer
134
views
Understanding proof of "if $f$ is non-negative, then it is the supremum of simple functions."
I am studying measure theory based on the book "Probability and Measure Theory" by Ash and have a question about understanding one of the theorems. Here it goes:
(Page 43) Theorem 1.5.9 (d) ...
0
votes
0
answers
62
views
Correct terminology about "probability measures" and "probability distributions"
INTRODUCTION. Let's consider some previous answers (from previous questions) about "probability measure" and "probablity distributions":
(Previous question 1) Distinguishing ...
0
votes
0
answers
32
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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 \...
2
votes
1
answer
71
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 ...
-1
votes
1
answer
56
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 ...
3
votes
0
answers
35
views
Accumulation in the zeroes of the rate function (Large deviation principle)
Consider a measure space $(\mathcal{X},\mathcal{B})$, where $\mathcal{X}$ is Polish and $\mathcal{B}$ is the Borel $\sigma$-field. Let $(\mu_n)$ be a sequence of probability measure that satisfies the ...
2
votes
0
answers
43
views
A lemma used to prove Besicovitch Theorem
Let $X$ be a metric space and $\mu\colon\mathcal{P}(X)\to[0,+\infty]$ an outer measure over $X$ such that all open subsets of $X$ are $\mu$-measurable (then $\mu$ is a Borel measure) and $\mu$ is ...
1
vote
0
answers
51
views
A question about Theorem 2.8.2 of "Geometric Measure Theory" of Federer.
Let $X$ be a metric space and $\mu\colon\mathcal{P}(X)\to[0,+\infty]$ an outer measure over $X$ such that all open subsets of $X$ are $\mu$-measurable (then $\mu$ is a Borel measure) and $\mu$ is ...
1
vote
1
answer
45
views
A question about Theorem 2.8.7 from "Geometric Measure Theory" of Federer
Let $X$ be a metric space and $\mu\colon\mathcal{P}(X)\to[0,+\infty]$ an outer measure over $X$ such that all open subsets of $X$ are $\mu$-measurable (then $\mu$ is a Borel measure) and $\mu$ is ...
4
votes
1
answer
130
views
Riesz-Markov theorem and positive linear functionals on real-valued continuous functions
Riesz-Markov theorem: Let $X$ be a locally compact Hausdorff space. For any continuous linear functional $\Psi$ on $C_0(X)$, there is a unique regular countably additive complex Borel measure $\mu$ on ...
1
vote
0
answers
43
views
If $|\mu(f^{-1}(U)) - \mu(g^{-1}(U))| < \epsilon$ for all $U$, do we have $\|f - g \circ \phi\|_1 < c(\epsilon)$ for some $\phi$?
Suppose $(\Omega, \mu)$ is a Borel probability space and $f, g : \Omega \to \mathbb{R}$ are measurable functions such that $|\mu(f^{-1}(U)) - \mu(g^{-1}(U))| < \epsilon$ for all Borel subsets $U \...
1
vote
1
answer
49
views
Hypothesis for establishing that the measure given by the Riesz representation theorem is a probability measure
I am trying to study, through the Riesz representation theorem applied to a space of compactly supported continuous functions, what hypotheses must be met to establish that the measure given by the ...
0
votes
0
answers
14
views
Sufficient conditions for switching the order of the limit and the lattice supremum for the family of weakly convergent sequences of measures.
Let $X$ be a metric space, complete and separable or even compact if needed. Fix some index set $I$, possibly uncountable. For each $i\in I$ consider a sequence $(\mu^i_n)$ of Borel probability ...
5
votes
1
answer
70
views
Does the weak convergence of probability measures imply some uniform over all sets estimates between these measures and the limit measure?
Let $X$ be a metric space. If it is necessary, one can assume that it is complete and separable, or even compact. Consider a sequence of Borel probability measures $(\mu_n)$ on $X$ that weakly ...
3
votes
1
answer
104
views
What is the cardinality of the set of measures $\mu$ on $(\mathbb{R}, \mathcal B (\mathbb{R}))$?
Having recently studied a course on $\mathsf{ZFC}$ Set Theory, I have been thinking about applications of the material that I have learnt in other disciplines of mathematics. With that in mind, I ...
0
votes
0
answers
34
views
monotone convergence of an Integral
I am currently working on the following task
Let $f \in \mathcal{M_+}(\mathbb{R},\overline{\mathbb{R}})$ an $\mu$ is a borel-measure on $\mathbb{R}.$ Show that,
$$\lim \limits_{n \to \infty} n \int_{-\...
2
votes
1
answer
58
views
Show that $\underset{E: \mu(E)<\delta}{\sup} \underset{j}{\sup} \displaystyle\int_E |f_j|d\mu \geq \epsilon.$
Let $f \in L_1(X, \mu)$. Show that for every $\epsilon > 0$ there exists $\delta > 0$ such that if $E \in M$ with $\mu(E) < \delta$, then
$$\int_E |f| d\mu < \epsilon.$$
Then, find $(X,\mu)...
2
votes
1
answer
93
views
Help with Theorem 2.9.7 from Federer's Geometric Measure Theory
Suppose $\phi$ and $\psi$ are Borel regular measures (outer measures) on a metric space $X$ such that $\phi(A),\psi(A)<\infty$ for every bounded subset $A\subseteq X$. One defines a Borel regular ...
0
votes
1
answer
57
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 ...
1
vote
0
answers
56
views
If $f,g$ are two measurable functions on $[0,1]$ such that $\int_A f=\int_A g$ for all $A\subset[0,1]$ such that $\mu(A)=1/3$. Then $f=g$ a.e. $\mu$
Here $\mu$ denotes the Lebesgue measure on $[0,1]$. WLOG, assume $g=0$. Then we are given with $$\int\limits_A f\ d\mu=0\ \forall A\subset[0,1]\text{ with }\mu(A)=1/3$$
First I assume that $f\ge0$. ...
2
votes
1
answer
99
views
About the definition of a Borel measurable function in "Measure, Integration & Real Analysis" by Sheldon Axler.
I am reading "Measure, Integration & Real Analysis" by Sheldon Axler.
2.35 Definition measurable function
Suppose $(X,\mathcal{S})$ is a measurable space. A function $f:X\to\mathbb{R}$ ...
0
votes
1
answer
37
views
Is $\mu_U(E) := \mu(E\cap U)$ a Radon measure when $U$ is open?
Let $X$ be a LCH space with a Radon measure $\mu$. By definition, Radon measure is a Borel measure, which is finite on compact sets, outer regular on Borel sets, and inner regular on open sets.
Now, ...
1
vote
1
answer
125
views
proove that $\forall x \in X : \forall \epsilon > 0 : \exists \delta > 0 \text{ s.t. } \mu(B(x,\delta)) ≤ \epsilon$
Im trying to solve the following exercise:
Let $X$ be a compact metric space and let $\mu$ be a finite Borel measure on $X$ such that
$\mu(\{x\}) = 0$ for all $x \in X$
(Here $B(x,\delta)$ denotes the ...
0
votes
0
answers
37
views
Borel measure - unbounded sets
Let $E\subset \mathbb{R}$ be borel-measurable and $\mu (E) < + \infty $, I have to show the following:
$$ \lim_{x \to +\infty} \mu ((E + \{ x \}) \cap E)=0 $$
A hint suggests to first consider ...
2
votes
1
answer
79
views
How can I prove that if $\int_{[0,1]} f~d\nu\leq C\int_{[0,1]} f~d\mu$ then the measures are absolutely continuous?
Let us consider $T:[0,1]\rightarrow [0,1]$ be a continuous map. Let $\mu, \nu$ be $T$ invariant probability measures on $\mathcal{B}([0,1])$. I want to show that if there exists $D>0$ s.t. $\int_{[...
4
votes
2
answers
218
views
Does a set of positive outer measure contain a *measurable* set of positive measure?
Given a complete measure space $(X, \mathcal{X}, \mu)$, and a subset $A \subseteq X$ of positive outer measure, does there necessarily exist a subset $E$ of a $A$ which is measurable for which $\mu(E) ...
0
votes
1
answer
38
views
regarding measurability of functions
in this question Borel Measurability of a function with countable discontinuity points. $X$ is a Borel set , but I have $2$ questions $1)$ if $X$ is every subset of $\mathbb{R}$ I think the theorem ...
0
votes
1
answer
29
views
Borel Measurability of a subset of $R^{\infty}$
Let $(a_m)_{m\geq 1}$ be a sequence of nonnegative real numbers and consider the set
$$
\Gamma=\left\{y\in\mathbb{R}^{\infty}:\sum_{m=1}^{\infty}a_my_m<\infty\right\}.
$$
I want to show that this ...
1
vote
0
answers
39
views
A detail on the Polar Coordinate Theorem proof
Assuming that Polar Coordinates Formula (Theorem 2.49 of Folland Real Analysis) works for simples functions, how to obtain the result for $L^1$ functions? That is, suppose that there exists an measure ...
3
votes
0
answers
46
views
Calculating a lower bound of Lebesgue measure of some Borel set
I have the following set. I need to calculate a lower bound on its measure in order to prove something about some distribution.
$I \subseteq [0,1)$. Take the set $\mathcal{J}(I) = I \bigcap \underset{\...