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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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.
Alberto's user avatar
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 ...
Oddly Asymmetric's user avatar
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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 ...
Rodrigo Palacios's user avatar
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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 ...
Minerva's user avatar
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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 ...
Biblioteca_da_medida's user avatar
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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 ...
NoetherBoy 's user avatar
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1 answer
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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 $\sigma$...
Euler007's user avatar
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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 $...
Euler007's user avatar
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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 $f_{\sharp}(\mu)$ ...
C L 's user avatar
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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, ...
Euler007's user avatar
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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. ...
Apple's user avatar
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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 ...
hteica's user avatar
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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}^...
Fran712's user avatar
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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, ...
AvengerHeld's user avatar
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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 ...
Jianing Song's user avatar
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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^...
Leonidas's user avatar
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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 $f : X \to [0;1]$ ...
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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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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 ...
JohnNash's user avatar
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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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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) ...
discretechoice's user avatar
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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 ...
Ommo's user avatar
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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 \...
OllyT777's user avatar
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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 ...
user754663's user avatar
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1 answer
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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 ...
Algo's user avatar
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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 ...
LUCA MAZZUCCO's user avatar
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0 answers
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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 ...
Grace53's user avatar
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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 ...
Grace53's user avatar
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1 answer
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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 ...
Grace53's user avatar
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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 ...
ayphyros's user avatar
1 vote
0 answers
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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 \...
Display name's user avatar
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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 ...
ayphyros's user avatar
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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 ...
Rafael's user avatar
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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 ...
Rafael's user avatar
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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 ...
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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_{-\...
WomBud's user avatar
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2 votes
1 answer
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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)...
Mr. Proof's user avatar
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2 votes
1 answer
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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 ...
dwhydtea's user avatar
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 ...
confusedTurtle's user avatar
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 $f\ge0$. ...
MathBS's user avatar
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2 votes
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 $f:X\to\mathbb{R}$ ...
tchappy ha's user avatar
  • 8,550
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, ...
Luke's user avatar
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1 vote
1 answer
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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 ...
user avatar
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 ...
undergradstudent123's user avatar
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_{[...
user1294729's user avatar
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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) ...
AJY's user avatar
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1 answer
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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 ...
Dsrksidemath's user avatar
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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 ...
Nicolas's user avatar
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1 vote
0 answers
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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 ...
Neymar Junior's user avatar
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{\...
user2582354's user avatar

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