# 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 ...
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### 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 ...
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### 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 ...
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### 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. ...
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### 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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### 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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### 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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### 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 ...
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### 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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### 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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### 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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### 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 ...
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### 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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### 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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### 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. ...
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### 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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### 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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### 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, ...
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### 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
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### 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
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### 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 $(X, d)$ ...
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### 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\...
### 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 ...