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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dual space of the subspace of the space of probability measures [closed]
I have a question which maybe so naive but I want to know the result about it.
Let $\mathcal{M}=\mathcal{M}(\mathbb{R})$ be the space of bounded measures. Then by some materiau such as ...
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Measurability of a 'cone'
Let A be a (Lebesgue) measurable set in $ \mathbb{R}^n$. Consider the 'cone with base A' $A(1) = \{\alpha x \in \mathbb{R}^n : x \in A, \alpha \in (0,1] \}$.
Is B Lebesgue measurable? I assume it is, ...
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Does equidistribution of zero average, due to irrationality, imply boundedness?
Let $f:\mathbb R\to\mathbb C$ be a sufficiently smooth and $1$-periodic function of average zero (i.e., $\int_0^{1}f(x)\,dx=0$), and let $\alpha\in(0,1)\smallsetminus\mathbb Q$. We know that
$$
\...
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On Radon measures with values in Banach space
It is known that continuous linear functionals on the space $C_0({\mathbb{R}^n})$ are bounded Radon measures ${\cal M}({\mathbb{R}^n})$ where $C_0({\mathbb{R}^n})$ is uniform closure of the space of ...
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Probability measures on $L^p$
Let $(X,\mathcal X,\mu)$ be a fixed measure space, and suppose that $\mu$ is stationary and ergodic with respect to the (left) action of a topological group $G$. Stationarity means that $\mu = g_* \mu ...
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Stone space of measure algebra [closed]
let $\lambda$ be the Lebesgue measure on the unit interval $I=[0,1]$, and $Leb(I)$ be the Boolean algebra of Lebesgue measurable in $I$ and $\mathcal{N}$ the family of Null sets. The measure algebra $\...
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Recovering measure from the family of sigma-subalgebras
Let $(X,\mathscr{F})$ be a measurable space.
Let $\mathscr{F}_\alpha \subseteq \mathscr{F}$, $\alpha \in \mathfrak{A}$ be a class of $\sigma$-subalgebras.
Let we have a measure $\mu_\alpha$ on every ...
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A argument related measurable partitions in dynamic system
$X$ is a compact metric space, and $T:X\rightarrow X$ be a continuous map, which is finite to one. Denoted by$ X_{0}$ the set of all points $x\in X$, such that for all sufficiently small $\epsilon>...
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When is a space of probability measures not perfectly normal?
I am looking for examples of pairs ($(\Omega,\Sigma)$, ($\mathcal P(\Omega)$, $\tau$)), where $(\Omega,\Sigma)$ is a measurable space and ($\mathcal P(\Omega)$, $\tau$) is a space of probability ...
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Are measurable functions almost surely constant on atoms?
Let $(\Omega,\Sigma,\mu)$ be a probability space. A $\mu$-atom is an $A\in\Sigma$ such that $\mu(A)>0$ and for all $B\in\Sigma$ such that $B\subseteq A$, either $\mu(B)=\mu(A)$ or $\mu(B)=0$ holds.
...
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Are there any good techniques for calculating Hausdorff measure?
I'm aware that many techniques have been developed for the purpose of calculating Hausdorff dimension (although I'm fairly unfamiliar with them), but my question is whether or not we have any good ...
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A non-trivial probability measure on $2^{\mathbb R}$
Consider the measurable space $2^{\mathbb R}$, equipped with the tensor-product $\sigma$-algebra. Famously, this space has a measurable structure which is not generated by a topology (see this answer)....
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functions of bounded variation and gradient vector measure
I want to prove a function of bounded variation on some domain $D\subset R^n$, $f\in BV(D)$, has the property that there is a constant $C$, such that
$$
\lim_{r\rightarrow 0}\frac{C}{r^{n+1}} \int_{...
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Entire functions with a null real escaping set
Let $f$ be a entire function (stable on $\mathbb{R}$), and $E_{\mathbb{R}}$ its real escaping set : $$E_{\mathbb{R}} = \{ x \in \mathbb{R} : f^{(k)}(x) \rightarrow_{k \to \infty} \infty \} $$
We put ...
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sub and super-levelset regularity for Sobolev functions
I'm wondering if there are known results about the "regularity" (in some sense to be determined) of sub and super levelsets of Sobolev functions $u\in W^{1,p}(\mathbb{R}^d)$. More precisely:
Assume $...
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a measure convolution equation
My question is:
Given a function $f$ in the Schwartz class, we are looking for a measure $\mu$ which is a solution of the convolution equation: $f = e^{-|.|^2/2} \ast \mu$, where $e^{-|.|^2/2}$ is ...
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Hausdorff measure and projections
Fix $ k \in \mathbb{N} $ and let $ H^k $ be the $k$-dimensional Hausdorff measure on $\ell^\infty $. Also, if $ V $ is a subspace of $ \ell^\infty $, we denote the projection onto $ V $ by $ \pi_V $. ...
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The First Failure of GCH in Large Cardinals Smaller than Measurables
A well known theorem by Scott says:
If $\kappa$ is a measurable cardinal and $\mu$ a normal measure on it and $\mu (\lbrace\lambda\in\kappa~|~2^{\lambda}=\lambda^{+}\rbrace)=1$ then $2^{\kappa}=\...
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Packing symmetric matrices in spectral norm, and defining measures on symmetric matrices
I'm trying to upper bound the $\epsilon$-packing number of $\Theta=\{A\in\mathbb{S}^{d}:\; a\preceq A \preceq b\}$ (where $\mathbb{S}$ are symmetric $d\times d$ matrices) for some $a\leq b$ with ...
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Forms of multivariate CLT [closed]
I am looking for a good reference for differnt kinds of multivariate central limit theorems. I was wondering how far the i.i.d. condition of the standard multivariate clt can be relaxed, as in can the ...
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When does $\ell_1(\Gamma)$ embed into $L_1(\mu)$?
Suppose we are given an uncountable set $\Gamma$ and a measure space $(\Omega, \mathcal{F}, \mu)$. I would like to know when the Banach space $\ell_1(\Gamma)$ embeds into $L_1(\mu)$. Of course, this ...
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Is sigma-additivity of Lebesgue measure deducible from ZF?
Is sigma-additivity (countable additivity) of Lebesgue measure (say on measurable subsets of the real line) deducible from the Zermelo-Fraenkel set theory (without the axiom of choice)?
Note 1. ...
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Finite measure on the power set
Let $X$ be an uncountable set, and let $\Omega$ be the power set of $X$, viewed as a $\sigma$-algebra. Does there exist a positive $\sigma$-additive measure of finite total mass on $(X, \Omega)$ such ...
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Differentiability of $f*g$ on the circle, for integrable f, bounded g, and some decay of the Fourier coefficients of f
If $f\in L^1(\mathbb{T})$ and $g\in L^\infty(\mathbb{T})$ where $\mathbb{T}$ is the circle, such that $\hat{f}\in L^{p}(\mathbb{Z})$ for some $1\leq p<\infty$, do we have that $f*g$ is ...
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strong convergence sufficient conditions
Would it be true that $\mu_n \to \mu$ strongly if $\int f\mathrm{d}\mu_{n}\to \int f\mathrm{d}\mu$ for every uniformly continuous function? Assume the space is $\mathbb{R}^{N}$ and has the usual ...
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Is there an inclusion of $L_\infty(G)$ into $C_0(G)^{**}$?
This is probably a silly question with which I am stuck however. Let $G$ be a locally compact group. It seems to me that there is a canonical map of $L_\infty(G)$ into $M=C_0(G)^{**}$ (the latter is ...
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Continuity on a measure one set versus measure one set of points of continuity
In short: If $f$ is continuous on a measure one set, is there a function $g=f$ a.e. such that a.e. point is a point of continuity of $g$?
Now more carefully, with some notation: Suppose $(X, d_X)$ ...
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What is the easiest way to show that three lines in two dimensional space do not intersect?
I have two similar questions:
1) Let $X$ and $Y$ be two measure spaces. Suppose for every point
$x \in X $ there exists a set $ \mathcal{U}_x \subset Y $ of full
measure in $Y$. Suppose $V \subset ...
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Conditional probabilities are measurable functions - when are they continuous?
Let $\Omega$ be a Banach space; for the sake of this post, we will take $\Omega = {\mathbb R}^2$, but I am more interested in the infinite dimensional setting. Take $\mathcal F$ to be the Borel $\...
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How to show that x-y is Lebesgue-Lebesgue measurable
Which is the cleanest way to show that the difference, $d:R^n\times R^n\rightarrow R^n$, $d(x,y)= x-y$, is Lebesgue-Lebesgue measurable? (i.e. foreach A lebesgue measurable in $R^n$, $d^{-1}(A)$ is ...
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Dual of the space of vector valued Borel measures
What is the dual of the space of all vector valued Borel measures?
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a convolutional equation for the gaussian measure [closed]
My question is: Let $\mu$ be the gaussian probability. Is there exists a measure $\nu$ solution of the equation:
$\nu*\mu=\delta$ where $\delta$ is the Dirac measure supported at the origin.
Thanks ...
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Is this a closed set?
Let $\Theta$ and $X$ be two (Hausdorff) topological spaces. Let $\mathbb P : \Theta \to \Delta(X)$ be a "statistical model", i.e., a continuous function from parameter space $\Theta$ to the space of ...
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A characterization of the module function on a locally compact division ring
The same question was asked in Math StackExchange about 3 months ago.
Since nobody has answered to it, I would like to post it here.
References:
Weil's Basic Number Theory(denoted by BNT).
Bourbaki'...
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Are measures of a measurable cardinal measurable? (Edited and Updated Version)
Update: Regarding to Prof. Hamkins's guidance I restricted the questions to the "normal" measures to avoid trivial answers.
Definition: Let $\kappa$ be a measurable cardinal. Define:
$\mathbb{M}_{\...
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A problem concerning measures on locally compact spaces
I am stuck on a question for quite sometime now, although in the text it is said to be "apparent". The problem goes as the following :
Let $X$ and $Y$ be locally compact Hausdorff spaces. Then $M(X)$ ...
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Product of Topological Measure Spaces
Def. A Radon measure $\mu$ on a compact Hausdorff space $X$ is uniformly regular if there is a countable family $\mathcal{A}$ of compact $G_\delta$-subsets of $X$ such that for every open set $U\...
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The facial structure of the convex hull of a family of characteristic functions
Let $S$ be a finite set and let $\mathcal{A} \subset\mathcal{P}(S)$ be a family of subsets of $S$. Consider the convex polytope spanned by the characteristic functions of members of $\mathcal{A}$ :
$$...
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Does the difference quotient of an absolut cont. funct. converge in L^1?
Assume that $\mu$ is a finite Radon measure on the real line and $f$ is integrable wrt. $\mu$. Define
$F(x)=\int_{]\infty;t]}f(y)d\mu(y) $
Is the following statement true?
The functions $d_h:x\...
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Extending a homeomorphism from a dense set [closed]
Let $X$ and $Y$ be Hausdorff topological spaces, and let $f : X \to Y$ be a Borel-measurable function. Suppose that $D \subseteq X$ is dense, that the image $f(D) \subseteq Y$ is dense, and that $f$ ...
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Can a compact metrizable space be determined by its Hausdorff measures?
Suppose that $(X,d)$ is a compact metric space. Now suppose that $h:[0,a]\rightarrow[0,b]$ is a continuous function with $h(0)=0$ where if $x\leq y$, then $h(x)\leq h(y)$. Then define $$L(d,h)=\lim_{\...
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Volume of fundamental domain and Haar measure
In my research, I do need to know the Haar measure. I have spent some time on this subject, understanding theoretical part of the Haar measure, i.e existence and uniqueness, Haar measure on quotient. ...
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Do semi-continuous functions generate bounded Borel measurable functions as a $C^*$-algebra?
This question is related to Question 2 of my previous posting.
Question. Let $\mu$ be a Radon measure on a compact Hausdorff space $\Omega$ and $L^{\infty}(\Omega,\mu)$ the set of essentially bounded ...
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Intersection of compact sets in the unit interval
Let $\mathscr K$ be an uncountable set such that every $K\in\mathscr K$ is a compact subset of $[0,1]$ with positive Lebesgue measure. Does it then follow that there exists an uncountable $\mathscr A\...
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Isometrically-invariant measures and dilation of the Cantor set
Let $C$ be the Cantor middle-thirds set. Let $\mu$ be a finitely-additive isometrically-invariant measure on all subsets of $\mathbb R$. Then $\mu(3C)=2\mu(C)$, where $aB = \{ ax : x \in B \}$. ...
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What conditions on a probability distribution defined by long-time averaging do I need to satisfy a central limit theorem?
For integer $n$, $1 \le n \le N$, consider the random variables
$X_n = \cos[t \omega_n]$
For any fixed $N$, we can take the mean
$Y_N = \frac{1}{N} \sum_{n=1}^N X_n$
and define a (cumulative) ...
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Uniform distribution in (non-compact) locally compact spaces
I'm trying to understand how much of the theory of uniformly distributed sequences in compact spaces can be extended to locally compact spaces.
Following L. Kuipers and H. Niederreiter - Uniform ...
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Original source for a well-known result of convergence in measure and almost everywhere
A well-known result in measure theory states that given a sequence $(f_n)_{n=1}^\infty$ of measurable functions from a $\sigma$-finite measure space $(X,\mathcal{A},\mu)$ to $\mathbb{R}$ then the ...
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Dual of $L^2(0,T,C)$ where $C'=BD(\Omega)$
Here is the hypothesis of my problem : $T>0$, $\Omega$ is a bounded open subset of $\mathbb{R}^n$ with a regular boundary. I'm actually looking for a proof that explains what is the dual of $L^2(0,...
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Determining the asymptotic behavior of random matrices with vanishing ratio dimensions
Consider an $N\times K$ random matrix $X$ (defined on a probability space $(Ω,F,μ)$) with i.i.d. entries having zero mean and variance $1/K$.
There are a lot of results regarding the asymptotic ...
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