Questions tagged [measure-theory]
Questions about abstract measure and Lebesgue integral theory. Also concerns such properties as measurability of maps and sets.
2,662
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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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+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. ...
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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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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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+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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1
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+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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1
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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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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
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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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143
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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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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 $\psi$:
$$\...
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79
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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 ...
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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 ...
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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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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}$, $\...
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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 ...
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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\...
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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 ...
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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 ...
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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}$$ ...
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545
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Analogue of open/closed maps for measurable spaces
$\newcommand{\A}{\mathcal{A}}\newcommand{\T}{\mathcal{T}}$The notions of continuous map of topological spaces and measurable function of measurable spaces are very similar:
A map of topological ...
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Does there exists a ball proximinal subspace in a $L_1$-predual space which is not an M-ideal
It has been proved that $M$-ideals in $L_1$-predual spaces are ball proximinal. But can we find a ball proximinal subspace that is not an $M$-ideal in an $L_1$-predual space?
I have taken $X=C[0,1]$ ...
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Interpretation of the Lévy measure of an infinitely divisible random vector
We know that a random vector $X$ is infinitely divisible (ID) if for all $n \in \mathbb N$, there exist $X_1^n,..., X_{n}^n$ i.i.d. random vectors such that:
\begin{equation}
X = X_1^n + ...+ X_n^...
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An infinite-dimensional linear programming problem related to Sobolev spaces
For any real $p>0$ and any Borel subset $B$ of the interval $I:=[0,1]$, let
$$\nu_p(B):=\int_0^\infty du\,pu^{p-1}\int_B dx\,g(u,x),$$
where $g\colon(0,\infty)\times I\to I$ is a measurable ...
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$\int_{\mathbb{R}}|p(v-r,x)-p(u-r,x)|\,dx \leq C\frac{v-u}{u-r}$
Consider $p(u,x)=(4\pi u)^{-d/2}e^{-\frac{|x|^2}{4u}},u>0,x\in \mathbb{R}^d.$
Prove that there exists $C>0$ such that for all $0<u\leq v,r\in[0,u[,$ $$\int_{\mathbb{R}^d}|p(v-r,x)-p(u-r,x)|\, ...
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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
$$f(x) \leq \mu\text{-...
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$\int_0^u\int_{[-1,1]^2}\int_{[-1,1]^2}\frac{1}{r}e^{-\alpha^2|x-y|^2/r} \, dx\,dy\,dr\leq Cu^{\epsilon}\alpha^{-2\beta}$
I am looking for a proof for the following fact: for $U>0,\beta>0,$ there exists $C>0,\epsilon>0$ such that $$\forall u\in [0,U],\alpha\in\left]0,1\right],\int_0^u\int_{[-1,1]^2}\int_{[-1,...
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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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2
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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
$$
u(x)...
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Open sets in the space of signed measures equipped with the Kantorovich–Rubinshtein norm
Let $X$ be a compact metric space and $\mathcal{M}(X)$ be the space of variational-bounded, signed Borel measures equipped with the Kantorovich–Rubinshtein norm, cf. [Section 8.3, 1]:
$$||\mu||_0:= \...
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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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1
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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 ...
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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 ...
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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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143
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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\...
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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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257
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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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1
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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 ...
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2
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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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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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