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

learn more… | top users | synonyms

5
votes
1answer
134 views

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 ...
7
votes
2answers
325 views

Measure theory in nuclear spaces

Much of the literature on measure theory in linear spaces focuses on the case of normed linear spaces (e.g., the outstanding book by Vakhania, or its sequel). However, nuclear linear spaces "as far ...
-2
votes
1answer
121 views

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 ...
25
votes
2answers
1k views

Is there a finite family of functions such that the max of any two functions can be dominated by a third?

Is it true that for every $t$ there is an $n$ and there exists a finite function family, $\cal F$, whose members are from $[n] \to \mathbb N$ (taking all different values) and for any $f_1, \ldots, ...
1
vote
1answer
115 views

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 $. ...
2
votes
0answers
62 views

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 ...
-2
votes
1answer
118 views

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 ...
3
votes
1answer
128 views

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 ...
4
votes
1answer
160 views

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 ...
10
votes
2answers
426 views

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. ...
6
votes
2answers
367 views

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 ...
1
vote
0answers
88 views

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 ...
13
votes
4answers
497 views

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 $(X, d_X)$ ...
0
votes
0answers
148 views

Dual of the space of vector valued Borel measures

What is the dual of the space of all vector valued Borel measures?
-1
votes
2answers
83 views

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 ...
2
votes
1answer
284 views

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 ...
2
votes
1answer
163 views

$X$ Polish geodesic implies $(P_2(X), W_2)$ geodesic

If $X,d$ is a complete and separable space then the space of Borel probability measures with finite second moment on $X$ endowed with the Wasserstein distance $W_2$ is geodesic. I am looking for ...
-1
votes
1answer
184 views

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). ...
2
votes
1answer
203 views

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: ...
1
vote
0answers
125 views

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 $M(X)$ ...
3
votes
2answers
175 views

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 ...
9
votes
0answers
319 views

Ultrafilter theorem and translation invariant measures

The usual Vitali construction of a non-Lebesgue measurable set generalizes to a proof that there are no (non-trivial) translation invariant measures on $\mathcal P\mathbb R$. On the other hand, there ...
1
vote
0answers
141 views

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 $f$ ...
1
vote
0answers
82 views

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 ...
22
votes
3answers
819 views

Does every set of reals contain a measure-zero set of the same cardinality? Does it contain a meager set of the same cardinality?

This question arises from an issue in my post on Ashutosh's excellent question on Restrictions of the null/meager ideal. Question 1. Does every set of reals contain a measure-zero subset of the same ...
1
vote
1answer
160 views

Lebesgue's integrability condition in several variables

The well known Lebesgue's condition of Riemann integrability says that a bounded function in one variable $f\colon [a,b] \to \mathbb{R}$ is Riemann integrable if and only if it is continuous almost ...
1
vote
0answers
145 views

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 ...
2
votes
1answer
163 views

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}} ...
2
votes
1answer
272 views

Fourier transforms of finitely additive bounded measures

Given a finitely additive positive regular bounded measure $\mu$ on ${\mathbb R}^n$ (i.e. a positive linear functional on $C_b({\mathbb R}^n)$), I wonder what can be said about its Fourier transform. ...
16
votes
2answers
512 views

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 ...
7
votes
2answers
164 views

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 $aB = \{ ax : x \in B \}$. ...
2
votes
1answer
209 views

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 ...
1
vote
1answer
133 views

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 ...
3
votes
2answers
596 views

Uniformly distributed sequence in $\mathbb{R}$

We say that a sequence $(x_n)_{n=1}^\infty \subseteq \mathbb{R}$ is "uniformly distributed in $[a,b]$", with $a < b$, if $(x_n)_{n=1}^\infty \cap [a,b] \neq \varnothing$ and $$\lim_{N \to \infty} ...
0
votes
1answer
399 views

What are some characterizations of the strong and total variation convergence topologies on measures?

I asked this question on StackExchange a few days ago but didn't get any response, so I thought I would try here. The Wikipedia article on convergence of measures defines three kinds of convergence: ...
3
votes
2answers
193 views

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 ...
2
votes
1answer
190 views

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 ...
17
votes
1answer
479 views

How strong is “all sets are Lebesgue Measurable” in weaker contexts than ZF?

Famously, Solovay showed that, if $\textrm{ZFC}$ plus $\textrm{IC}$ (the existence of an inaccessible cardinal) is consistent, then so is $\textrm{ZF}$ plus $\textrm{DC}$ (dependent choice) plus ...
2
votes
0answers
223 views

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 ...
8
votes
3answers
314 views

reference request : constructive measure theory

As the title said, I would like to know if constructive measure theory has been developed somewhere ? I am more precisely interested in the (constructive) theory of completely continuous valuation on ...
2
votes
0answers
156 views

An inequality for Lp-functions

I am interested in the following inequality: \begin{equation} \int\limits_\Omega \left\vert f_1 - f_2 \right\vert^p ~ d\mu \le C_p \left[ \int\limits_\Omega \left\vert f_1 \right\vert^p ~ d\mu + ...
3
votes
0answers
66 views

A small planar set containing a large family of curves

A beautiful construction described in [1] shows a compact connected plane set of measure zero containing circles (circumferences) of every radius between zero and one. A corollary to a theorem proved ...
7
votes
0answers
146 views

Qualitative weakenings of probabilistic independence

In probability theory, independence of random variables is characterised by $$(1)~~X~\text{independent}~Y \; \iff \; P_{(X,Y)} = P_X \otimes P_Y \enspace ,$$ where $P_{(X,Y)}$ is the joint probability ...
1
vote
1answer
91 views

Submodular measures on the hypercube

By the hypercube I mean the lattice formed by all n-bit strings ordered by pointwise inequality. For example, $000 \leq 110$, $010 \leq 110$, $110$ and $001$ are not comparable. Further we have the ...
6
votes
0answers
295 views

Prokhorov's theorem for finite signed measures?

Prokhorov theorem provides a useful characterization of relatively compact sets w.r.t. narrow topology (topology induced by narrow convergence) in the space of probability measure. Notation used ...
2
votes
1answer
164 views

Empirical estimator fot the total variation distance on a finite space

I have two probability measures $p$ and $p'$ on a finite set $X$ which I do not know precisely, but which I can sample from. I would like to estimate their total variation (omitting multiplier $2$): ...
3
votes
1answer
131 views

Weak continuity of Lebesgue decomposition

Let $X$ be a space with its $\sigma$-algebra $\mathcal{B}$; we are given a finite measure $\mu$ and a sequence of finite measures $\nu_n$ such that, for every bounded continuous function ...
4
votes
1answer
190 views

Doubt on Morrey spaces of measures according to T. Giga and Y. Miyakawa

I'm reading 'Navier-Stokes flow in $\mathbb{R}^3$ with measures as initial vorticity and Morrey spaces' (a paper of Giga and Miyakawa) but there is something that I don't understand about the ...
2
votes
0answers
86 views

convolution of surface measures

Thanks for reading my question. Assume we are given two compact (possibly with boundary) $(n-1)$-dimensional $C^{\infty}$ manifolds $M_1$ and $M_2$ embedded in $\mathbb{R}^n$, with induced measure ...
1
vote
1answer
234 views

A set of positive-measure not being a countable union of cylinder sets and zero-measure sets?

Let $(A^\mathbb{N}, \mathcal{B}(A^\mathbb{N}), \mu)$ be a measure space, where $A^\mathbb{N}$ is a set of one-sided sequences over a finite alphabet $A \subset \mathbb{N}$, $\mathcal{B}(A^\mathbb{N})$ ...