Questions tagged [measure-theory]

Questions about abstract measure and Lebesgue integral theory. Also concerns such properties as measurability of maps and sets.

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Name for a regularity property of $\sigma$-ideals

Let $X$ be a topological space and let $\mathcal{B}$ be its Borel $\sigma$-algebra. Suppose $\mathcal{N} \subset \mathcal{P}(X)$ is a $\sigma$-ideal, i.e. $\emptyset \in \mathcal{N}$ and it is closed ...
1 vote
1 answer
112 views

Question on density of certain set of matrices

Let $B$ be an invertible real matrix and let $Q=\{A \text{ real}\mid AB^{T} \text{ is symmetric}\}$. Is the subset $S=\{ A \in Q\mid A+A(B^{-1}A)^{2} \text{ is symmetric}\}$ of measure zero in $Q$? I ...
1 vote
0 answers
40 views

Regularity of $\sigma$-finite measure pushforwarded by completion

Let $(X, d)$ be a metric space. Let $\mu$ be a $\sigma$-finite measure defined on borel subsets of $X$. Let $i \colon X \to \hat{X}$ be an isometry on image, where $\hat{X}$ is a complete metric space ...
4 votes
1 answer
200 views
+50

Integrating on orbits of algebraic groups

Suppose $G$ is a $\mathbb{Q}$-algebraic group (I am interested in the semisimple case) acting rationally on a vector space $V_\mathbb{Q}$. Let $x \in V_\mathbb{Q}$ be a non-zero rational vector. ...
3 votes
0 answers
69 views

Question on an integral inequality

I am reading van de Vaart and Weller, Weak Convergence and Empirical Processes With Applications to Statistics. And I am stuck in the proof of Theorem 2.6.7 on page 141. For simplicity I restae the ...
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1 vote
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Sets of Hausdorff measures zero

Let $H^g$ be the Hausdorff measure with respect to gauge function $g$. I need to construct an example of a set E for which: $H^g(E)=0$ for $g(r)=r^s$, $s>0$ and $0<H^g(E)<+\infty$ for $g(r)=2^...
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2 votes
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+100

Closed graph correspondence which never contains the whole support

Let $I=[a,b]$ with $a<b\in\mathbb{R}$ and denote by $\mathcal{M}(I)$ the set of Borel probability measures on $I$ equipped with the topology induced by the weak convergence of measures. Does there ...
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2 votes
1 answer
166 views
+200

Identity for spectral resolution: $dE_{\xi, \xi}= |g|^2 dE_\eta, \eta{}$

Let $(\Omega, \mathcal{F})$ be a measurable space. Let $E: \mathcal{F}\to B(H)$ be a regular resolution of the identity on the Hilbert space $H$, see e.g. Rudin's functional analysis book. Suppose ...
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2 votes
1 answer
38 views

p.d.f. of exponential family

I have a question about the p.d.f. of exponential family. Suppose $(X,\mathcal{F})$ is a measurable space and $\{F_{\theta},\theta\in \Theta\}$ is a distribution family on $(X,\mathcal{F})$. When $\{...
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1 vote
1 answer
97 views

How to show that $\int x \,d\nu = 0$ using a pseudo-weak convergence of measures?

I have a sequence of $p$-dimensional infinitely divisible random vectors $S_n'$, such that $S_n' \Longrightarrow X$, as $n \to \infty$. Suppose the following assumptions The characteristic functions ...
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46 views

Does weak convergence of filtrations preserve progressive measurability?

Suppose on some probability space $(\Omega, \mathcal{F}, \mathbb{P})$ I have a sequence of filtrations $\mathbb{F}^n =(\mathcal{F}^n_t)_{t \geq 0}$ generated by Brownian motions $W^n$ for each $n$, ...
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25 views

Prove the explicit form of the ratio function in a Markov Chain

Let $(A_M^{\mathbb{Z}_+}, \Omega, P, \lambda)$ be a Markov Shift where $A$ is a finite alphabet set, $M$ is the admissibility matrix, $P = [P_{i, j}]_{i, j\in A}$ is a stochastic matrix that is ...
6 votes
1 answer
483 views

Integration in Banach algebra

Let $\mu$ be a Borel measure on the real line $\mathbb{R}$ taking values in a separable Banach algebra $A$. Assume that $\mu$ is such that the total variation measure $|\mu|$ is finite. Let $f$ be a ...
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3 votes
1 answer
143 views

Examples of convergence in distribution not implying convergence in moments

It is well know that the convergence in distributions does not necessarily imply convergence in expectation, but implies convergence in expectation of bounded continuous functions. Let $\{X_n\}$ be a ...
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5 votes
1 answer
104 views

Kernel of bounded operator $C_0(\mathbb{R})\to C_0(\mathbb{R})$

Let $T:C_0(\mathbb{R})\to C_0(\mathbb{R})$ be a bounded linear operator, where $C_0(\mathbb{R})$ is the space of continuous functions on the real line vanishing at the infinity equipped with the ...
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78 views

Approximation arguments

I am a student reading about Luo and Sarnak's paper and I have trouble understanding the conclusion. In the paper this theorem is proved for a continuous function of compact support $\psi$: $$\...
1 vote
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79 views

About vector valued measure algebras

Let $G$ be a locally compact group and $A$ be a Banach algebra. By $L^1(G,A)$ and $M(G,A)$ we denote the $A$-valued group, and measure algebra. Is $M(G,A)$ a Banach algebra (with convolution as the ...
-1 votes
0 answers
73 views

Is there a specific term for measure spaces that do not have a metric, or that only allow a diagnonal metric?

In the physical sciences, we often work in spaces that have a natural notion of area or volume, but that lack a metric. It is common to work with probability density functions or point process ...
1 vote
2 answers
70 views

A limit definition of regular conditional probability

I am working with a proof that would greatly benefit from a definition of conditional probability along the lines of the obscure unreferenced alternative definition found in Wikipedia. A Wikipedia ...
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2 votes
0 answers
59 views

A type of coupling problem II

This posting is related the following questions in MSE and in MO. In general terms, suppose $(X,\mathscr{B},\mu)$ is a $\sigma$-finite measure space. If $\nu$ is another measure on $\mathscr{B}$, $\...
2 votes
1 answer
63 views

Measurability of random function with values in $C(K,E)$

Let $K \subset \mathbb{R}$ be compact, and let $E$ be a separable Banach space. Further, let $(\Omega, \mathcal{F},\mathbb{P})$ a probability space. I would like to show that a certain a random ...
3 votes
1 answer
131 views

A type of coupling problem I

This posting is related to a recent question asked in MSE: Suppose $(X,\mathscr{B},\mu)$ is a $\sigma$-finite measure space. If $\nu$ is another measure on $\mathscr{B}$, $\nu(X)=\mu(X)$, and $\nu\ll\...
3 votes
1 answer
65 views

Vague Topologies induced by $C_c$ and $C_0$ are the same on a closed ball of finite Radon measures?

Let $X$ be a locally compact Hausdorff space. Denote $C_c(X)$ and $C_0(X)$ the space of continuous functions with compact support and vanishing at infinity respectively. By Riesz representation ...
1 vote
1 answer
97 views

Equivalence of unions in probability theory

I am working with Ash's Probability and Measure Theory, Second Edition, specifically on theorem 6.2.1 (some convergence criteria for a random variable sequence). We are given a sequence $(X_i)_{i \ge ...
2 votes
1 answer
239 views

Strongly uniform infinite binary strings

For $A\subseteq \omega$ we let the lower and upper density be defined as $$\mu^-(A):= \lim\inf_{n\to\infty}\frac{|A\cap n|}{n+1} \text{ and } \mu^+(A):= \lim\sup_{n\to\infty}\frac{|A\cap n|}{n+1}$$ ...
9 votes
2 answers
545 views

Analogue of open/closed maps for measurable spaces

$\newcommand{\A}{\mathcal{A}}\newcommand{\T}{\mathcal{T}}$The notions of continuous map of topological spaces and measurable function of measurable spaces are very similar: A map of topological ...
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0 votes
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23 views

Does there exists a ball proximinal subspace in a $L_1$-predual space which is not an M-ideal

It has been proved that $M$-ideals in $L_1$-predual spaces are ball proximinal. But can we find a ball proximinal subspace that is not an $M$-ideal in an $L_1$-predual space? I have taken $X=C[0,1]$ ...
1 vote
0 answers
55 views

Interpretation of the Lévy measure of an infinitely divisible random vector

We know that a random vector $X$ is infinitely divisible (ID) if for all $n \in \mathbb N$, there exist $X_1^n,..., X_{n}^n$ i.i.d. random vectors such that: \begin{equation} X = X_1^n + ...+ X_n^...
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25 views

An infinite-dimensional linear programming problem related to Sobolev spaces

For any real $p>0$ and any Borel subset $B$ of the interval $I:=[0,1]$, let $$\nu_p(B):=\int_0^\infty du\,pu^{p-1}\int_B dx\,g(u,x),$$ where $g\colon(0,\infty)\times I\to I$ is a measurable ...
0 votes
1 answer
65 views

$\int_{\mathbb{R}}|p(v-r,x)-p(u-r,x)|\,dx \leq C\frac{v-u}{u-r}$

Consider $p(u,x)=(4\pi u)^{-d/2}e^{-\frac{|x|^2}{4u}},u>0,x\in \mathbb{R}^d.$ Prove that there exists $C>0$ such that for all $0<u\leq v,r\in[0,u[,$ $$\int_{\mathbb{R}^d}|p(v-r,x)-p(u-r,x)|\, ...
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0 votes
1 answer
94 views

An inequality involving the essential supremum

Let $\mu$ be a Radon measure on $[0, 1]$, and $f: [0, 1] \to \mathbb R$ a Borel measurable function. Question: Is it true that for $\mu$ almost every $x \in [0, 1]$, we have $$f(x) \leq \mu\text{-...
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2 votes
1 answer
139 views

$\int_0^u\int_{[-1,1]^2}\int_{[-1,1]^2}\frac{1}{r}e^{-\alpha^2|x-y|^2/r} \, dx\,dy\,dr\leq Cu^{\epsilon}\alpha^{-2\beta}$

I am looking for a proof for the following fact: for $U>0,\beta>0,$ there exists $C>0,\epsilon>0$ such that $$\forall u\in [0,U],\alpha\in\left]0,1\right],\int_0^u\int_{[-1,1]^2}\int_{[-1,...
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7 votes
0 answers
128 views

Relationship between measure theory and quantification

I was advised that this question might be better suited for mathoverflow, so I am reposting it here (original post). In a 1978 paper published by David P. Ellerman and Gian-Carlo Rota, the duo discuss ...
1 vote
1 answer
169 views

Inequality and integral

Let $p(u,x):=(4 \pi u)^{-1/2}e^{-\frac{x^2}{4u}},u>0,x \in \mathbb{R}.$ Let $\mathcal{E}:=\{\phi \in C_c^\infty (\mathbb{R}),\operatorname{supp}(\phi) \subset B(0,1),\|\phi\|_\infty \leq 1\}.$ ...
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1 vote
1 answer
70 views

Prove that $N_1(A_1,A_2)= N_2(A_1,A_2)$ when $A_1A_2=A_2A_1$ and $A_1$ and $A_2$ are normal operators

Let $\mathcal{L}(E)$ be the algebra of all bounded linear operators on a complex Hilbert space $E$. On $\mathcal{L}(E)^2$, we have two equivalent norms: \begin{eqnarray*} N_1(A_1,A_2) &=&\sup\...
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0 votes
0 answers
62 views

What is compatibility?

This is rather subjective. But when we say "a measure is compatible with the topology" what do we mean exactly? Disclaimer: I'm not being sarcastic. I'm not being mathematically hostile. ...
0 votes
1 answer
94 views

Integral and inequality

Let $p(u,x):=(4 \pi u)^{-1/2}e^{-\frac{x^2}{4u}},u>0,x \in \mathbb{R}.$ Let $\mathcal{E}:=\{\phi \in C_c^\infty (\mathbb{R}),\operatorname{supp}(\phi) \subset B(0,1),\|\phi\|_\infty \leq 1\}.$ ...
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0 votes
1 answer
220 views

Integral with inequality

Let $p(u,x):=(4 \pi u)^{-1/2}e^{-\frac{x^2}{4u}},u>0,x \in \mathbb{R}.$ Let $\mathcal{E}:=\{\phi \in C_c^\infty (\mathbb{R}),\operatorname{supp}(\phi) \subset B(0,1),\|\phi\|_\infty \leq 1\}.$ ...
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4 votes
2 answers
150 views

Lebesgue differentiation theorem at boundary points for Sobolev traces

$\newcommand{\R}{\mathbb R}$ Let $\Omega\subset \R^d$ be a smooth, bounded open set and fix $p\geq 1$. Fact 1: the usual Lebesgue differentiation theorem says that, if $u\in L^p(\Omega)$, then $$ u(x)...
2 votes
1 answer
147 views

Open sets in the space of signed measures equipped with the Kantorovich–Rubinshtein norm

Let $X$ be a compact metric space and $\mathcal{M}(X)$ be the space of variational-bounded, signed Borel measures equipped with the Kantorovich–Rubinshtein norm, cf. [Section 8.3, 1]: $$||\mu||_0:= \...
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1 vote
0 answers
33 views

Measurability in a product space of a set defined only along its fibers

Consider the probability space $([0,1],\mathcal{B}([0,1]),\lambda)$, where $\mathcal{B}([0,1])$ denotes the Borel $\sigma$-algebra in $[0,1]$ and $\lambda$ is the Lebesgue measure in $[0,1]$. Then, ...
2 votes
1 answer
193 views

Measuring how "close" $\alpha\in[0,1]\setminus\mathbb{Q}$ is to being rational

Let $\mathbb{N}_+$ denote the set of positive integers and let $\mathbb{N}_0 = \mathbb{N}_+\cup\{0\}$. Fix $\alpha\in[0,1]\setminus \mathbb{Q}$. For $n\in\mathbb{N}_+$ we let the approximation radius ...
1 vote
2 answers
81 views

Measurability of Brjuno numbers

A positive irrational number $\alpha\in{\mathbb R}\setminus {\mathbb Q}$ is said to be a Brjuno number if $$\sum_{i=1}^\infty\frac{\log q_{i+1}}{q_i} < \infty$$ where $q_i>0$ is the denominator ...
1 vote
0 answers
37 views

Dynamical formulation of the 2-Wasserstein distance for *discrete* matrix-valued measures

TL;DR: I want to find a definition generalizing "$t \mapsto \frac{1}{m} \sum_{k = 1}^{m} \delta_{x_k(t)}$ is a Wasserstein gradient flow" to matrix-valued probability measures. Let $(X, d)$ ...
0 votes
0 answers
143 views

Remainder-balancedness of primes

Let $\mathbb{N}_+$ denote the set of positive integers. Consider the remainder function $\text{rem}:\mathbb{N}_+\times \mathbb{N}_+ \to \mathbb{N}\cup\{0\}$ defined by $$(n,d) \mapsto n - \Big(\Big\...
0 votes
1 answer
97 views

Can we further restrict the space of test functions to $C_c^\infty (X)$ in weak convergence?

Let $X := \mathbb R^n$, $C_b(X)$ the space of all real-valued bounded continuous, $C_c(X)$ the space of all real-valued continuous functions with compact supports, and $C_c^\infty(X)$ the space of ...
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6 votes
1 answer
257 views

A characterisation of continuous real functions

Let $f: \mathbb R^n \to \mathbb R$ be a measurable function. We say $f$ is precise if for every $x \in \mathbb R^n$ and every compact subset $K$ of $\mathbb R^n$ such that for $|K \cap B_\delta (x)|&...
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0 votes
1 answer
81 views

Sums of powers of measures of $p$-adic balls

Let $(a_n,k_n) \in \mathbb{Z}_p \times \mathbb{N}$ for $n \in \mathbb{N}$ and consider the sequence of closed $p$-adic balls $B(a_n,k_n) = a_n + p^{k_n}\mathbb{Z}_p$. I assume that the $(a_n,k_n)$ are ...
6 votes
2 answers
148 views

Bisector of two points in a Riemannian manifold has measure $0$

Let $p,q\in M$, $p\neq q$, where $M$ is a Riemannian manifold. We will let the bisector of $p,q$ be $\mathcal{B}(p,q)=\{x\in M;d(p,x)=d(q,x)\}$. Does $\mathcal{B}(p,q)$ have measure $0$? I was ...
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6 votes
2 answers
127 views

When can we extend a function on a $\lambda$-system to a probability measure?

Let $\Omega$ be a nonempty set and let $\mathcal{L}$ a $\lambda$-system on $\Omega$. That is, (i) $\Omega \in \mathcal{L}$, (ii) if $A, B \in \mathcal{L}$ and $A \subseteq B$, then $B \setminus A \in \...

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