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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Complement of the support of measure on R^n has measure 0

How do we prove that $\mu (B) = 0$ for any ball $B$ such that $B \subset S^c$? For any $x \in S^c$, there exists $r_x > 0$ such that $\mu \left( B(x, r_x) \right) = 0$. Why there exists a countable ...
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Whether or not a measure is finite or $\sigma$-finite

I have been given the following question: Let $\mu$ be a Borel measure on $[1, \infty)$ given by the density function 1/x with respect to the 1-dimensional Lebesgue measure. Is measure $\mu$ finite or ...
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Let $f: \mathbb R → \mathbb R_+$ and $g:\mathbb R →\mathbb R_+$ be two Borel measurable functions. Show that: [closed]

I have been given the following question: "Let $f: \mathbb R → \mathbb R_+$ and $g:\mathbb R →\mathbb R_+$ be two Borel measurable functions. Show that: $$(∫_\mathbb R f(x)g(x)/x^2dλ(x))^2 ≤ ∫_\...
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Weak Compactness Theorem for Borel Measures

Weak convergence is defined as follows: The sequence $\{\mu_j\}$ of Borel measures on $\Bbb R^n$ converges weakly to a Borel measure $\mu$ if for all $f\in C_0(\Bbb R^n)$, $$\int f d\mu_j \to \int f ...
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Help understanding the definition of a specific measure.

In an exercise, I am supposed to determine if a measure is finite or $\sigma$-finite, however, I do not understand the definition of said measure. The measure is defined as a "Borel measure on $[...
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84 views

Show that $S^{\otimes T}=\underset{I \,is\, a\, countable\, subset\, of\, T}{\cup}\,\,\,\pi_1^{-1}(S^{\otimes I})$

Let $S$ be a measurable space and $T$ an uncountable index set. For a subset $I\in T$, we write $\pi_1:S^T\to S^I$ for the natural projection. Show that $S^{\otimes T}=\underset{I \,is\, a\, countable\...
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Understanding proof that $f$ is Borel measurable.

Consider the function $f:\mathbb{R}\rightarrow\mathbb{R}$ defined such that $\forall x\leq y :f(x)\leq f(y)$. My lecture notes state that $f$ is Borel measurable with the seemingly simple proof: $$\...
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Equivalence relation induced by a free action of an infinite countable group is aperiodic?

I am reading on how to calculate the cost of direct products in measurable group theory. In the proof by Gaboriau we have the following: Let $\Gamma, \Delta$ be infinite countable groups. We want to ...
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How is this version of Portmanteau theorem well-defined?

Let $(X, d)$ be a metric space and $\mathcal P(X)$ the space of all Borel probability measures on $X$. I'm reading below theorem in this lecture note. Lemma 6.2. Suppose $\mu, \mu_1, \mu_2,\ldots \in ...
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Can a complex Radon measure be approximated by compactly supported Radon measures?

Let $G$ be an (abelian) locally compact Hausdorff group. Consider the following fragment from Folland's text "A course in abstract harmonic analysis" (second edition, p102). Why is the ...
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If $X$ is complete separable, then the space $\mathcal{P}(X)$ of all Borel probability measures on $X$ is separable in Prokhorov metric

I'm trying to prove below result. Could you verify if my attempt is fine? Let $(X, d)$ be a metric space and $\mathcal{P} :=\mathcal{P}(X)$ the space of all Borel probability measures on $X$. Let $d_P$...
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Example of a sequence of Borel measures that converges weakly, but does not converge in Lévy–Prokhorov metric

Let $(X, d)$ be a metric space and $\mathcal P(X)$ the space of all Borel probability measures on $X$. We endow $\mathcal P(X)$ with the Lévy–Prokhorov metric $d_P$. Let $\mathcal C_b(X)$ be the space ...
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If either $\mathcal M(X)$ or $\mathcal P(X)$ is complete, then so is $X$

I'm trying to prove this result from Wikipedia. Could you have a check on my attempt? Let $(X, d)$ be a metric space. Let $\mathcal{M} :=\mathcal{M}(X)$ and $\mathcal{P} :=\mathcal{P}(X)$ be the set ...
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Compute explicitly Lévy–Prokhorov metric for $2$ finite Dirac measures

Let $(X, d)$ be a metric space and $\mathcal{M} :=\mathcal{M}(X)$ the space all non-negative finite Borel measures on $X$. The Prokhorov metric $d_P$ on $\mathcal{M}$ is defined by $$ d_{P}(\mu, \nu) :...
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The space of Borel probability measures is closed in that of finite Borel measures w.r.t. Lévy–Prokhorov metric

I'm trying to prove this intuitive result. Could you have a check on my attempt? Let $(X, d)$ be a metric space. Let $\mathcal{M} :=\mathcal{M}(X)$ and $\mathcal{P} :=\mathcal{P}(X)$ be the sets all ...
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A sufficient condition for a collection of Borel probability measures to be uniformly tight

I'm trying to prove below result. Could you verify if my attempt is fine? Let $(X, d)$ be a complete metric space and $\mathcal{P}(X)$ the space of Borel probability measures on $X$. Let $\Gamma \...
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Every finite Borel measure on a separable metric space is tight.

I'm trying to prove this property of Polish space. Could you verify if my attempt is fine? Let $(X, d)$ be a complete separable metric space. Then every finite Borel measure on $X$ is tight. I post ...
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Let $\mu(X) = \infty$. Is the collection of Borel sets on which $\mu$ is both outer and inner regular a $\sigma$-algebra?

Let X be a Hausdorff topological space $X$ and $\mathcal B(X)$ its Borel $\sigma$-algebra. Let $\mu$ be a non-negative Borel measure and $B \in\mathcal B(X)$. $\mu$ is tight on $B$ iff $$\mu(B) = \...
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2 votes
2 answers
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Riesz-Markov theorem: any positive linear functional is continuous?

Below is the Riesz–Markov theorem that I take from here. Riesz–Markov theorem: Let $X$ be a locally compact Hausdorff space and $C_0(X)$ the space of continuous compactly supported functionals on $X$....
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The underlying metric space is separable if and only if the space of finite Borel measures is separable in Prokhorov metric

I'm trying to prove below result about Prokhorov metric. Could you verify if my attempt is fine? Let $(X, d)$ be a metric space and $\mathcal{M} :=\mathcal{M}(X)$ the set all non-negative finite ...
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If the underlying metric space is separable, then weak convergence is equivalent to convergence in Prokhorov metric

I'm rewriting the proof that weak convergence is equivalent to convergence in Prokhorov metric in separable metric space. Could you verify if my attempt is fine? Let $(X, d)$ be a metric space and $\...
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2 votes
1 answer
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Prokhorov metric can be extended to all finite Borel measures?

I'm reading about Prokhorov metric, i.e., Let $(X, d)$ be a metric space and $\mathcal{P} :=\mathcal{P}(X)$ the set all Borel probability measures on $X$. Let $$ d_{P}(\mu, \nu) := \inf \left\{ \...
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Triangle inequality of Prokhorov metric

I'm trying to prove that Prokhorov metric satisfies triangle inequality, i.e., Let $(X, d)$ be a metric space and $\mathcal{P}(X)$ the set all Borel probability measures on $X$. Let $$ d_{P}(\mu, \nu)...
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If $\mu_i(A) \to \mu(A)$ for all $A$ with $\mu(\partial A) = 0$, then $\int_E g \mathrm d \mu_i \to \int_E g \mathrm d \mu$

I'm trying to prove below equivalence of weak convergence of finite Borel measures. Let $(E, d)$ be a metric space and $\mu, \mu_1, \mu_2,\ldots$ finite Borel measures on $E$. Let $g:E \to \mathbb R$ ...
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approximating a measure by smooth measures

Let $M\subset \mathbb R^n$ be a compact $C^\infty$ manifold with boundary $S\in C^\infty$ with a surface measure $ds$ induced from $\mathbb R^n$ and $\eta$ is a non negative finite Borel measure on $...
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2 votes
1 answer
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Significance of Luzin's theorem.

I am studying measure theory from Axler's book.I am a graduate student.So,it is essential to understand the essence of each theorem.I should not be satisfied by understanding the proof only.Now,there ...
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1 vote
1 answer
105 views

Fréchet derivative of the total variation norm for measures on a manifold

Let $\Theta$ be a compact $d$-dimensional Riemannian manifold without boundary and $M(\Theta)$ (resp. $M_+(\Theta)$) denote the set of signed (resp. nonnegative) finite Borel measures on $\Theta$. ...
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Constructing a measure on a product space of ordered sets

Let $X$ and $Y$ be linearly ordered sets equipped with their interval topologies. Let $\mu$ and $\nu$ be finite positive Borel measures on $X$ and $Y$, respectively. If $I ⊆ X$ and $J ⊆ Y$ are each ...
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1 answer
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Sigma algebra properties exercise

Could you prove that the sigma algebra σ generated by the classes $A_1$ and $A_2$ satisfy the condition:$ σ(A_1\cup A_2)=σ(σ(A_1)\cup\sigma(A_2))$ I've been trying to do this exercise but I'm not been ...
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2 votes
0 answers
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Optimal transport map between Lebesgue and Borel measures

Let $\mu$ be the Lebesgue measure on $[0, 1]$ and $\nu$ be the Borel measure defined on $[0, 1]$ by $$\int_{[0, 1]} f(y) \nu(dy) = (1 - \alpha) \int_{0}^{1} f(y) dy + \alpha f(1) ~~~~~ \forall f \in \...
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How is a restricted measure defined over sets not in sub-sigma algebra?

According to ProofWiki given a measure space $(\mathsf{X}, \mathcal{X}, \mu)$ and a subsigma algebra $\mathcal{Y}\subseteq\mathcal{X}$, the restriction of $\mu$ to $\mathcal{Y}$ is the measure $\mu\...
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Bijection $h:[0,1)\rightarrow [0,1)^{\mathbb{N}}$

I would like to find a referenceable source of the following cool technique to get an explicit bijection $h:[0,1)\rightarrow [0,1)^{\mathbb{N}}$ that is measurable and has measurable inverse: Cool ...
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Show that the length of the union of intervals is less than (or equal to) the sum of the lengths of each of those intervals.

Let's define a generalised interval $\langle a,b \rangle$. $\langle a,b\rangle$ could be either one of the following: $[a,b], (a,b), [a,b), (a,b]$. Now let's define a length function on these ...
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3 votes
1 answer
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Show that $\mu \geq 0$ on Borel subsets of $X.$

Let $X$ be a compact Hausdorff space and $\mu$ be a complex Borel measure on $X$ such that $\int_{X} f\ d\mu \geq 0$ for every $f \in C(X)$ with $f \geq 0.$ Then show that $\mu$ is non-negative. ...
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6 votes
1 answer
213 views

Is there a measurable function from $[0,1]$ to $ω_1$?

Does there exist a measurable function from $[0,1]$ (with the Lebesgue measure) to $ω_1$ that induces the Dieudonné measure? Definitions: $ω_1$ is the set of all countable ordinals, equipped with its ...
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How is the measure of an arbitrary set calculated?

I am following Folland's Real Analysis text to learn measure theory and have so far been able to understand how Borel measures are constructed over $\mathbb{R}$ from increasing, right continuous ...
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1 vote
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Compactness in Kantorovich Duality Problem

I've been following along in https://lchizat.github.io/files2020ot/lecture1.pdf to learn about optimal transport theory and ran into some confusion in Chapter 4 "The dual problem"... Let $X,...
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2 votes
1 answer
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$f$ is $\mathbb{B}$ measurable if and only if $g$ is $\mathbb{B}$ measurable and $P,N \in \mathbb{B}$

Let $\mathbb{B}$ be a $\sigma$-algebra on a set $X$ and let $f\,:\,X \rightarrow \overline R$ be an extended real-valued function. Define the sets $P\:= f^{-1}(-\infty)$ and $N\:=f^{-1}(+\infty)$. ...
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1 vote
1 answer
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Show that $f \in L_\infty(\mathbb{R})$ as soon as $f'$ is a Radon measure of finite total variation.

Let $f$ be a generalized function in $\mathcal{D}'(\mathbb{R})$ such that its derivative $\mathrm{D} f = \mu \in \mathcal{M}(\mathbb{R})$, the space of signed Radon measure with finite total variation,...
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Proof of the fact that every continuous function is Borel measurable.

I am doing a course on Measure theory in my Masters' course.I try to do proofs on my own whenever possible. Definition A function $f:X\subset \mathbb {R\to R}$ is called Borel measurable if the ...
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1 vote
0 answers
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A reference to a theorem about L1-bounded Martingales

I'm auditing a course called "Topics in Analysis", and in class we mentioned a theorem that didn't make sense to me. I don't fully remember how it goes, but it was something like: Theorem (...
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2 votes
1 answer
102 views

A measure is completely determined by its values on borel sets?

Let us assume that we have two probability measures defined on a compact set $X\subset \mathbb{R}^d$ with a $\sigma$-algebra which properly contains Borel sets of X (considering the topology given by ...
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1 vote
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The (upper/lower) derivative of a locally finite Borel measure w.r.t. another such measure is a Borel-function.

Let $\mu$ and $\lambda$ be locally finite Borel measures on $\mathbb{R}^n$. We may define an upper resp. lower derivative of $\mu$ w.r.t $\lambda$ at $x\in\mathbb{R}^n$ by $$ \underline{D}(\mu,\lambda,...
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2 votes
0 answers
56 views

Atoms for a regular Borel measure on a second countable, locally compact Hausdorff space.

Suppose that $X$ is a locally compact Hausdorff space, and let $\mu$ be a positive measure defined on the $σ$-algebra of Borel subsets of $X$, with $μ(X)<∞$. Definition 1. One says that $μ$ is ...
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1 vote
1 answer
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Proving a function is in $L^{\infty}$

Let $f:\Bbb{R^2}\to\Bbb{R}$ be a lebesgue measurable function, $f\in L^{\infty}$. For every $\theta\in [0,2\pi)$, define $g_\theta(x)=f(x\cos(\theta),x\sin(\theta))$. Prove that for almost every $\...
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1 vote
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Measure of closed set is vaguely u.s.c. on M_1^+(K)

I'm working my way through Compact convex sets and boundary integrals by Alfsen and during the proof of Proposition I.2.8 they use that set $W=\{\mu\in M_1^+(K)|\mu(\overline{V})<\alpha\}$ is ...
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1 vote
0 answers
40 views

Sets in completed measure space equals additive sets

I‘m trying to prove that the sets in the completed (finite) measure space $(X,\mathcal{E},\mu)$ are the additive sets. Definitions: We have defined a set $A\in\mathcal{P}(X)$ as additive if: $$\mu^{*}(...
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5 votes
2 answers
116 views

Is a measure measurable?

This could totally be a stupid question but I'm unsure: is a measure (ie positive, countable additive on a $\sigma$ algebra, 0 for the empty set) actually a measurable function (wrt to the Borel-sigma ...
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2 votes
1 answer
61 views

The Lebesgue $\sigma$-algebra $L(\mathbb{R}^n)$ is the completion of the Borel $\sigma$-algebra $B(\mathbb{R}^n)$

I came across the following proof of the completion of Borel $\sigma$-algebra to $\sigma$-algebra comprised of Lebesgue measurable sets which I cannot understand quite clearly. Can anyone elaborate ...
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7 votes
3 answers
218 views

Constructing the Haar measure of the $n$-dimensional Torus

Let $\mathbb{T}^n:=\mathbb{R}^n/\mathbb{Z}^n$ be the quotient of the group $(\mathbb{R}^n,+)$ by the subgroup $(\mathbb{Z}^n,+)$. I'm trying to construct the Haar measure of $\mathbb{T}^n$. I ...
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