**2**

votes

**1**answer

69 views

### 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, ...

**2**

votes

**1**answer

135 views

### Rate of convergence of the average of an equidistributed sequence

Let $f : \mathbb R\to\mathbb C$ be an $1$-periodic and sufficiently smooth function, which has zero average, and let $\alpha$ irrational. We know the following:
a. ...

**2**

votes

**1**answer

162 views

### Proving that Brownian motion has no points of increase

I am reading Burdzy's paper on the points of increase of Brownian motion:
Burdzy's Paper
He is proving that, almost surely, a Brownian motion, has no points of increase. What he actually proves is ...

**4**

votes

**1**answer

235 views

### 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
$$
...

**1**

vote

**0**answers

109 views

### Uniform Law Of Iterated Logarithm for VC classes

Kenneth Alexander proved a uniform Law Of Iterated logarithm for Vapnik-Chervonenkis classes in the article Probability Inequalities for Empirical Processes and a Law of the Iterated Logarithm (Ann. ...

**3**

votes

**1**answer

145 views

### 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 ...

**2**

votes

**1**answer

195 views

### 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 ...

**0**

votes

**1**answer

139 views

### 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 ...

**1**

vote

**0**answers

82 views

### 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 ...

**2**

votes

**1**answer

135 views

### 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 ...

**3**

votes

**1**answer

156 views

### 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.
...

**1**

vote

**0**answers

79 views

### 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 ...

**2**

votes

**1**answer

132 views

### 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 ...

**3**

votes

**1**answer

175 views

### Particular neighborhoods of analytical sets

Let $X$ be a standard Borel space: a topological space isomorphic to a Borel subset of a complete separable metric space. Denote by $\mathcal P(X)$ the set of all Borel probability measures over $X$ ...

**9**

votes

**3**answers

567 views

### measure with given push-forwards

Let $X,Y$ be locally compact spaces (in my specific case, they are locally compact groups). Suppose that we are given a measure $\mu$ on $X$ and a finite number of quotient maps ...

**4**

votes

**3**answers

371 views

### 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 ...

**5**

votes

**1**answer

131 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

**2**answers

305 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

**1**answer

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

**2**answers

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

**1**answer

110 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

**0**answers

61 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

**1**answer

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

**1**answer

116 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

**1**answer

158 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

**2**answers

376 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

**2**answers

360 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

**0**answers

86 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

**4**answers

485 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

**0**answers

126 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

**2**answers

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

**1**answer

280 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

**1**answer

158 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

**1**answer

182 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

**1**answer

197 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

**0**answers

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

**2**answers

168 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

**0**answers

309 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

**0**answers

137 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

**0**answers

81 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

**3**answers

790 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

**1**answer

151 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

**0**answers

137 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

**1**answer

156 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

**2**answers

265 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

**2**answers

500 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

**2**answers

153 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

**1**answer

200 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

**1**answer

131 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

**2**answers

595 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} ...