Questions tagged [measure-theory]
Questions about abstract measure and Lebesgue integral theory. Also concerns such properties as measurability of maps and sets.
3,071 questions
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Heine-Borel property for (probability) measures on $\mathcal{S}'$?
For Euclidean space $\mathbb{R}^n$, it is a well-known fact that the Schwartz space $\mathcal{S}(\mathbb{R}^n)$ is a Montel space. Moreover, the space of tempered distributions $\mathcal{S}'(\mathbb{R}...
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Why do we need to define a random variable as a function?
I recently learned the mathematical definition of a random variable, namely:
A random variable is a measurable function $X: \Omega \rightarrow \mathbb{R}$ whose domain $\Omega$ is equipped with a $\...
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Ensuring the measure condition $\mu(E) = \lambda$ in a lemma: need some clarification regarding the selection of $A$
I was studying a lemma from my notes on ergodic theory and encountered a difficulty. The lemma states:
Let $(X, \mathcal{B}, \mu)$ be an infinite non-atomic measure space, and let $T$ be an ergodic ...
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Comonotone solution for Optimal Transport problems with supermodular surplus
In Alfred Galichon's book Optimal Transport Methods in Economics the foollowing result is stated for OT problems on the real line.
Theorem 4.3.(i) Assume that $\Phi$ is supermodular. Then the primal ...
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Does $\int_{\mathbb R^d} (1+|x|^{1 + \alpha}) \ell (x) \, d x < \infty$ imply $\int_{\mathbb R^d} (1+|x|) |\ell (x)|^{1-\alpha} \, d x < \infty$?
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We fix $\alpha \in (0, 1)$. Let $\ell : \bR^d \to \bR_+$ be a continuous function such that
$$
\|...
- 114k
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Lower bound on the Lévy-Prokhorov metric for normal distributions
Let $\mathfrak M(\mathbb R^n)$ denote the metric space of probability measures (over $\mathbb R^n)$ equipped with the Lévy-Prokhorov metric $\rho$. Consider two $n$-variate normal distributions $\...
- 128k
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Golomb subsets of $\mathbb{N}$
A set $A\subseteq\newcommand{\N}{\mathbb{N}}\N$ is said to be Golomb if whenever $a<b \in A$ and $a'<b' \in A$ with $(b-a) = (b' - a')$, then $a=a'$ and $b=b'$. If $A\subseteq \N$ is Golomb, we ...
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Intuition for why Kelly criterion is 'so simple'
The Kelly criterion gives a simple formula to calculate the fraction of one's current wealth/bankroll.
On the page above it says
Assuming that the expected returns are known, the Kelly criterion ...
- 12.9k
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Does the sequence of bounded symmetric square integrable holomorphic functions have a convergent subsequence?
Let $f$ be a bounded holomorphic function on $\mathbb D^2$ and $s : \mathbb C^2 \longrightarrow \mathbb C^2$ be the symmetrization map given by $s(z) = (z_1 + z_2, z_1 z_2),$ for $z = (z_1, z_2) \in \...
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Majorization theory on $\sigma$-finite measure spaces
I want to learn about majorization and submajorization theory on $\sigma$-finite measure spaces. I know things get a bit more complicated compared with the case of a finite measure spaces but I'm ...
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Law of large numbers for a continuum of Bernoullis
Suppose I have a family of $n$ independent Bernoulli random variables described by a vector of parameters $(p_i)_{i=1}^n$. As it is well known, the number of successes within this family is a random ...
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Signed measures on algebras (fields) and their boundedness properties
I asked this question here on math.StackEchange, but it might be too technical so I re-post it here.
Let $X$ be a compact Hausdorff second countable topological space. Let $\mathcal{B}$ a countable ...
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How decreasing can a bijection $f:\mathbb{N}\to\mathbb{N}$ be?
This is a follow-up to this question by
Dominic van der Zypen. For each bijection $f:\mathbb{N}\to\mathbb{N}$, let
$$\operatorname{rc}(f) := \liminf_{N\to\infty} \frac{\left|\left\{(m,n)\in\{1,\dots,N\...
- 114k
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Existence of sequence of regular projections
Reading the book :Krasnosel'skii, M.A.; Pustylnik, E.I.; Sobolevskii, P.E.; Zabreiko, P.P. (1976), Integral Operators in Spaces of Summable Functions, Leyden: Noordhoff International Publishing, 520 p....
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Is projection of a closed subspace Borel?
Specifically, letting $E$ be a separable infinite-dimensional real Banach space, and $D_2$ in $E\times E$ a closed linear subspace, is then $\{\,x:\exists\,y\,;(x,y)\in D_2\}$ a Borel set in $E\,$? ...
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Min–max reversing bijections $f:\mathbb{N}\to\mathbb{N}$
For any set $X$, let $\newcommand{\N}{\mathbb{N}}[X]^2 = \big\{\{x,y\}:x\neq y \in X\big\}$ and set $[n]^2 = [\{0,\dotsc,n-1\}]^2$ for any positive integer $n$. For $A\subseteq [\N]^2$ we set $$\...
- 12.9k
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Modifiying a sequence of measures to assign a certain value when integrating a fixed function?
Let $f:\mathbb R ^d\to \mathbb R$ be some continuous function, $|f|\leq A(1+|x|)$, where $|\cdot|$ denotes the usual Euclidean norm. Fix a measure $\mu$ and constant $C$.
Assume that $\mu_n$ is a ...
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Sufficient condition for uniform convergence of the Stieltjes transform
Let $\mu$ be a probability measure and $\mu_N$ be a sequence of probability measures. For simplicity we may assume them to have compact supports contained in $[-1,1]$. Define
$$G_\mu(z):=\int\frac{\mu(...
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Sequential compactness of a sequence of curves of Borel probability measures
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Sparse "bijection-proof" subsets of $[\mathbb{N}]^2$
We call a collection ${\cal S}\subseteq {\cal P}(\newcommand{\N}{\mathbb{N}}\N)$ bijection-proof if for any bijection $\varphi:\N\to\N$ there is $T\in{\cal S}$ with $\varphi(T) \in {\cal S}$.
For any ...
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Is there an equivalent condition for Borel projections being Borel?
Let $X,Y$ be Polish spaces, and $B\subseteq X \times Y$ a Borel subset. The projection $B_X$ is not necessarily Borel in $X$. I have seen a few sufficient conditions for the projection to be Borel, ...
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Extreme points of a set of probability measures
Consider the set of Borel-measurable probability measures over the interval $[0,1]$ with a given mean, say 1/2. To be precise, I'm talking about the following set $$M=\left(\mu\in \Delta([0,1]):\int x\...
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Lebesgue measure of the boundary of the positivity set of a function is zero?
Let $w$ be a function $\mathbb R^n\to \mathbb R$ with the following properties:
$w$ is globally $\alpha$-Hölder continuous, $\alpha \in (0,1)$;
$w$ is biharmonic on $C=\{w>0\}$;
$w$ is subharmonic ...
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Stability of Hölder constants of frozen Itô stochastic integrals
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Shrinking and expanding pairs in bijections $\varphi:\mathbb{N}\to\mathbb{N}$
Motivation. If we consider any bijection $\varphi:\newcommand{\N}{\mathbb{N}} \N \to \N$, we say integers $m\neq n$ are shrinking with respect to $\varphi$ if $|m-n|>|\varphi(m) - \varphi(n)|$, and ...
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How to check that the surface measure is the weak limit of $\delta^{-1}\mathcal{L}^n|_{B(0,1+\delta)\setminus B(0,1)}$?
We denote the unit sphere $\{x\in\mathbb{R}^n:|x|=1\}$ by $S^{n-1}.$ If $x\in\mathbb{R}^n\setminus\{0\}$, the polar coordinates of $x$ are
\begin{align*}
r=|x|\in(0,\infty),\quad \gamma=\dfrac x{|...
- 128k
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Is there $\varepsilon \in (0, 1)$ such that $\sup_{t \in [0, \varepsilon]} [\ell_t]_\beta < \infty$?
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On representing a continuous time Markov chain by a stochastic integral of a Poisson random measure
Let $Q=(q_{ij})$ be the transition rate matrix of a continuous time Markov chain $\{ X_t \}$ with countable state space $M$. Let $q_i = -q_{ii}=\sum_{j \neq i}q_{ij}$, and let $\Gamma_{ij}$ be defined ...
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How can the maximal ideal space of the Fourier Stieltjes algebra be non-separable?
I have been asking a fair few (probably elementary) questions about abstract harmonic analysis lately. By means of explanation, I am just feeling around the subject at the moment and trying to build ...
- 1,074
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567
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Darboux property of non-atomic sigma-additive nonnegative measures equivalent to the AC?
A result commonly, and probably erroneously, attributed to W. Sierpiński is that every non-atomic, countably additive, nonnegative measure $\mu: \Sigma \to \bf R$, where $\Sigma$ is a sigma-algebra on ...
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Uncountable collections of distinct subsets of an interval (existence)
Throughout, $\mu$ is just the Lebesgue measure.
Question: does there exist an uncountable family of distinct subsets of $[-1, 1]$, denoted by $(U_j)_{j \in [-1, 1]}$, with $\mu(U_j) > 0$ for each $...
- 12.9k
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Gaussian Hypercontractivity of Chaos based on Gaussian with value in Hilbert spaces?
The classical Gaussian hypercontractivity is stated as following: Suppose $\xi$ is a Gaussian variable and $H_n(\xi)$ is the space of n-th homogeneous Wiener chaos constructed from $\xi$, then for any ...
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Billingsley convergence of probability measures - inequality used in Theorem 2
On Page 8, Billingsley defines $f(x)=(1-\rho(x,F)/\epsilon)^{+}$ where $\rho(x,F)$ is the metric distance from the set $F$. He then states $|f(x)-f(y)|\leq \rho(x,y)/\epsilon$ and goes on to use this ...
CommunityBot
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Sufficient conditions for the space of Radon measure to be a Banach space
Let $\mathcal{X}$ be a Hausdorff space and consider the space of Radon measures with bounded total variation $M(\mathcal{X})$ on $\mathcal{X}$.
Usually, the additional assumptions on $\mathcal{X}$ are ...
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How wild is the maximal ideal space of the Fourier-Stieltjes algebra of the real line?
The Fourier-Stieltjes algebra of $\mathbb R$ is the set of all sufficiently nice measures on $\mathbb R$. The vector product is convolution of measures. By identifying each measure with its Fourier ...
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Topological measure theory on spaces that are not completely regular
In the usual discourse regarding approaches to measure theory, it is often pointed out that the restriction of topological measure theory to locally compact Hausdorff spaces is insufficient. However, ...
- 3,816
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Approximate a non-negative function which is measurable in product $\sigma$-algebra
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Let $D$ be the set of those $\omega \in \Omega$ such that $f(\omega, \cdot)$ is $\mu$-integrable. Is $D$ measurable?
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Equality of two measures on functional spaces
It is well known that if $\mu$ and $\nu$ are two measures on the space $C^0([0,1],\mathbb{R}^n)$ of continuous mappings from $[0,1]$ to $\mathbb{R^n}$ endowed with its Borel $\sigma$-algebra satisfy $$...
CommunityBot
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Approximation on $H^1_0(B)$ and cut-off functions
Let $u \in H^1_0(B)$, where $B$ is the unit ball in $\mathbb{R}^N$. Given $\epsilon > 0$, I need to show there exists a function $\chi_\epsilon \in C^\infty_0(\mathbb{R}^N)$ such that
$$
\| u - \...
- 2,670
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About the sigma algebra generated by the Hausdorff measure on $\mathbb R^n$
Let $\mathcal{H}^k$ be the $k-$dimensional Hausdorff measure on $\mathbb R^n$, with $k \in \{1, \ldots n\}$. By Carathéodory's theorem we know that there exists a sigma algebra $\mu(\mathcal{H}^k)$ of ...
CommunityBot
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Does the Doob-Dynkin lemma hold for any measurable space that separates points?
This is cross-posted from math.se, where I got no responses.
Let's say that a measurable space $(Z, \mathcal Z)$ has the "Doob-Dynkin property" iff for any set $X$, measurable space $(Y, \mathcal Y)$ ...
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729
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Probabilty measures that are both discrete and continuous
Consider a measure space $\left(S,\Sigma\right)$ where each state $s\in S$ can be expressed as $s=\left(x,c\right)$, where $x\in\mathbb R$ and $c\in\mathbb N$. E.g., suppose $s$ denotes the state of a ...
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If $f=h\circ g$, then there's a measurable function $\tilde h$ such that $f=\tilde h\circ g$
Let $X,Y,Z$ be three standard measurable spaces and $f:X\to Z$ and $g:X\to Y$ two measurable functions. Suppose that there's a function $h:Y\to Z$ such that $f=h\circ g$. How can I show that there's a ...
- 285
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Let $A:=\{f\in C^1(\mathbb{R}): \hat{f}, \hat{f'} \in L^1(\mathbb{R})\}$. Schwartz space is dense in $A$ wrt $\|f\|:= \|\hat{f}\|_1+\|\hat{f'}\|_1$?
Let $A:=\{f\in C^1(\mathbb{R}): \hat{f}, \hat{f'} \in L^1(\mathbb{R})\}$, where $\hat{f}$ is the Fourier transform of $f$. Then is it true that Schwartz space $\mathcal{S}(\mathbb{R})$ is dense in $A$ ...
- 12.9k
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Is a $\sigma$-algebra generated by complete independent $\sigma$-algebras also complete?
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Let $(\Omega, \cA, \mu)$ be a probability space and $\cA_1, \cA_2$ sub $\sigma$-algebras of $\cA$. Let $\...
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Is it true that $F(X_0, \cdot) = X_0 + \int_0^T \sigma(s, X_0) \, \mathrm d B_s$ a.s.?
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Is $F: \mathbb T \times \mathbb R^d \times \Omega \to \mathbb R^d$ (constructed from Itô integral) Borel measurable in the product $\sigma$-algebra?
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Continuity of disintegrations in non locally compact spaces
Let $X$ and $Y$ be Radon spaces, $\mu$ a Borel probability measure on $X$, $F\colon X\to Y$ measurable. Then the disintegration theorem gives us a disintegration $\{\mu^y\}_{y\in Y}$ of $\mu$ with ...
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construction of a random measure with a given mean
Let me first pose a trivial question.
Given a Borel probability measure $\mu$ on the real line, is it possible to construct a purely atomic random measure $M$ whose mean is $\mu$?
The answer is ...
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