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
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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.
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2
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1
answer
51
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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 ...
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0
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23
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$\mu_0$ (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 ...
- 327
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0
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48
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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 ...
- 125
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0
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25
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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
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0
answers
64
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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
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1
answer
40
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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 $\sigma$...
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2
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0
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58
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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 $...
- 125
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0
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21
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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 $f_{\sharp}(\mu)$ ...
- 149
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1
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56
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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,
...
- 125
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1
answer
15
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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. ...
- 79
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0
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36
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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 ...
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1
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0
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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}^...
- 311
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2
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71
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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
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1
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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 ...
- 1,318
1
vote
0
answers
25
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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^...
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2
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89
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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 $f : X \to [0;1]$ ...
- 636
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1
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27
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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 $\...
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65
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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 ...
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1
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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 ...
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2
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1
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134
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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
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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 ...
- 319
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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 \...
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2
votes
1
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71
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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 ...
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1
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56
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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 ...
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3
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0
answers
35
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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
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0
answers
43
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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 ...
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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 ...
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1
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45
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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 ...
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1
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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 ...
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1
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0
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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 \...
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1
answer
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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 ...
- 75
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0
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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 ...
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5
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1
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70
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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 ...
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3
votes
1
answer
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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 ...
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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_{-\...
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1
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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)...
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1
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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 ...
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1
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57
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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
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0
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56
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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 $f\ge0$. ...
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2
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1
answer
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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 $f:X\to\mathbb{R}$ ...
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1
answer
37
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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, ...
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vote
1
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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 ...
user1178244
0
votes
0
answers
37
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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
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1
answer
79
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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_{[...
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2
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218
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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) ...
- 8,639
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1
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38
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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 ...
- 383
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votes
1
answer
29
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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 ...
- 157
1
vote
0
answers
39
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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 ...
- 891
3
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0
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46
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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{\...
- 94