**10**

votes

**1**answer

336 views

### Every measure on a set $X$ extends to the power set of $X$: Consistent or not with ZF?

Question. Is it consistent with ZF that every (countably additive, non-negative) measure $\mu: \Sigma \to \bf R$, where $\Sigma$ is a sigma-algebra on a given set $X$, extends to a (countably ...

**4**

votes

**1**answer

121 views

### Weak convergence in random measures

I don't understand the following as I read along a proof in a paper (Page 66, "Asymptotic Behaviour of some interacting systems", by Sylvie Meleard):
We denote by $\mathcal{P}({M})$ the space of ...

**2**

votes

**1**answer

94 views

### Do we have independence if we let the indices of the events increase?

Let $(\Omega, \mathscr F, \mathbb P)$ be a probability space.
Consider events indexed by $m, n \in \mathbb N$:
$ \ \ \ \ \ \ \ \ \ \ \ A_{1,n}, A_{2,n}, A_{3,n} ...$ are n-wise independent.
...

**2**

votes

**1**answer

138 views

### Extremally disconnected spaces and a measure theoretic property

Suppose that $X$ is an extremally disconnected topological space (meaning that the closure of an open set is still open). Then $X$ has the following property: the family of all sets $S$ such that $S$ ...

**4**

votes

**1**answer

82 views

### Question about the weak convergence of probability

Let $\mu$ be a probability measure on $\mathbb R$ and set
$$c(K):=\int_{\mathbb R}(x-K)^+d\mu(x).$$
Assume that one has a sequence of probability measures $(\mu_n)_{n\ge 1}$ s.t.
$$\int_{\mathbb ...

**1**

vote

**1**answer

115 views

### Question abouth Skorokhod representation of random variables (II)

This is a continuation of
Question abouth Skorokhod representation of random variables
Let $\mu$ and $\nu$ be two probability measures on $\mathbb R$ such that
$$\int_{\mathbb R}|x|^pd\mu(x),~ ...

**5**

votes

**1**answer

109 views

### Question abouth Prokhorov metric

Let $X$ and $Y$ be two random variables with first order moments, i.e. $E[|X|]$, $E[|Y|]<+\infty$. Assume further that
$$E\left[|X-Y|\right]<\varepsilon.$$
Set $Law(X)=\mu$ and $Law(Y)=\nu$, ...

**3**

votes

**0**answers

156 views

### The projection of density $1$ point on a rectifiable set

I posted this question on MSE but I received no reply. So I repost this here for better luck. Thank you!
Let $\Gamma\subset \mathbb R^N$ be $\mathcal H^{N-1}$-rectifiable. Then we know that $\mathcal ...

**4**

votes

**1**answer

130 views

### Is there a mixing condition to get the decay property I want?

Let $(X,\mu)$ be a probability measure space and $T:X\to X$ an ergodic invertible measure preserving transformation.
Consider a measurable set $A\subset X$ with $0<\mu(A)<1$
For each $N$ define ...

**2**

votes

**0**answers

50 views

### Doubts regarding pre-compactness of bounded sequence of measure valued functions

Given:
$\phi(x,\lambda) : \Omega$X$\mathbb R \to \mathbb R^{n}$ be a Caratheodory vector such that for each $M \gt 0 $, $\alpha_{M}(x) = max_{|u| \leq M} |\phi(x,u)| \in L^{2}_{loc} (\Omega)$ .
...

**3**

votes

**1**answer

155 views

### Question abouth Skorokhod representation of random variables

It is known that for any two probability measures $\mu$ and $\nu$ on $\mathbb R$ that are close in the Prokhorov metric $\rho$, i.e.
$$\rho(\mu,\nu)<\varepsilon,$$
then there exist two random ...

**7**

votes

**0**answers

247 views

### Generator of a $\bigoplus_{n=0}^\infty \mathbb{Z}/2\mathbb{Z}$-action

Let $T$ be a measure-preserving action of a group $G$ on a Lebesgue space $X$. That means that $T$ associates an automorphism (i.e. an invertible measure-preserving transformation) $T^g$ of $X$ to ...

**0**

votes

**0**answers

67 views

### $\sum_{n\in\mathbb{Z}^2}d\mu(x-2\pi n)=0\Rightarrow$ the summands are pairwise mutually singular

Let $\mu$ a finite measure supported by $\Gamma$ (smooth curve in $\mathbb{R}^2$) and absolutely continuos with respect to the arc length measure on $\Gamma$.
Please why if ...

**0**

votes

**1**answer

164 views

### $\int_{R^2}\varphi(x)d\mu(x)=0$ $\Leftrightarrow$ $\sum_{n\in \mathbb Z^2} d\mu(x-2\pi n)=0$

Let $\mu$ be a finite measure supported by $\Gamma $ (a smoth finite curve) and absolutely continuous with respect to the length measure on $\Gamma$ such that $\Gamma \cap (\Gamma+x)$ is a finite ...

**6**

votes

**1**answer

121 views

### Generator determined by finitely many translates implies zero entropy

Let $T$ be a measure preserving transformation of a standard probability space $(X,\mathcal{B},\mu)$. A partition $\alpha$ of $X$ is said to be a generator for $T$ if the smallest $T$ invariant ...

**1**

vote

**0**answers

139 views

### Sigma algebra generated by SOT versus of sigma algebra generated by WOT

Let $H$ be a non-separable Hilbert space. Let us denote $B_s$ ($B_w$), by the sigma algebra generated by the strong operator topology (weak operator topology) on $B(H)$.
Question: Is $B_s$ the same ...

**0**

votes

**1**answer

187 views

### Is the limsup or liminf of n-wise independent events independent?

Let $(\Omega, \mathscr F, \mathbb P)$ be a probability space.
Consider events indexed by $m, n \in \mathbb N$:
$ \ \ \ \ \ \ \ \ \ \ \ A_{1,n}, A_{2,n}, A_{3,n} ...$ are n-wise independent.
...

**1**

vote

**1**answer

84 views

### Calculate correlation values of an ensemble of $N\times N$ real asymmetric random matrix from Gaussian measure

I am now reading a paper by Sommers, H. J., et al. "Spectrum of large random asymmetric matrices." Physical Review Letters 60.19 (1988): 1895-1898., it claims a mathematical statement (equation (2) in ...

**8**

votes

**1**answer

570 views

### Sort-of Converse of Kolmogorov Zero-One Theorem

Let $(\Omega, \mathscr F, \mathbb P)$ be a probability space. The Kolmogorov Zero-One Theorem states that
Suppose we have independent random variables $X_1, X_2, ...$. Then $\forall \ A \in ...

**6**

votes

**1**answer

183 views

### When is Hausdorff measure a Frostman measure?

Let $(X,d)$ be a metric space and let $\mathcal{H}^s$ be the $s$-dimensional Hausdorff measure on $X$.
For a measure $\mu$ on $X$, we say that $\mu$ is a Frostman measure (sometimes referred as ...

**4**

votes

**1**answer

180 views

### Operator-valued measurable functions

Let $H$ be a non separable Hilbert space and $\Omega$ be a measurable space.
Naturally, we say that $f:\Omega\to B(H)$ is $w$-measurable if $f^{-1}(O)$ is measurable for any open set $O$ in the weak ...

**3**

votes

**1**answer

96 views

### Number of lattice points in homotetic image

I asked this question on MSE a week ago and it gave me a tumbleweed badge :-)
Let $\Lambda$ be a lattice in $\mathbb R^n$, with covolume $\Gamma$.
Moreover, let $S$ be a bounded (Lebesgue-)measurable ...

**2**

votes

**0**answers

75 views

### Moment Sequence in l²

I have the following problem/question:
For which finite regular complex measures $\mu$ is the moment sequence
$$
\left(\int_{[-1,1]}t^k\,d\mu\right)_{k\in\mathbb N}
$$
a member of $\ell^2(\mathbb ...

**3**

votes

**0**answers

80 views

### What are universal abstract $\sigma$-algebras on $\sigma$-frames?

Originally asked on MSE.
In this paper, the authors make the following definitions:
An (abstract) $\sigma$-algebra is a boolean algebra with countable joins.
A $\sigma$-frame is a bounded lattice ...

**2**

votes

**0**answers

141 views

### On the use of the term “field of sets” in Maharam's papers

I am reading some papers by D. Maharam, and feel a little bit confused about her use of the term "field of sets". Nowadays, I think the term is standardly used to mean a pair $(X, \mathscr{F})$ for ...

**6**

votes

**1**answer

187 views

### 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 ...

**3**

votes

**0**answers

67 views

### Stochastic equation

Let $X,Y$ be Polish spaces and $\kappa:X\times \mathcal B(Y)\to[0,1]$ be a Borel-measurable stochastic kernel on $Y$ given $X$. Under which conditions for a probability measure $\nu$ on $Y$ there ...

**1**

vote

**1**answer

143 views

### A boundary of the second fundamental theorem of calculus

Let's say that a set $X\subseteq [0,1]$ has Property Q if the following holds: For every continuous $f:[0,1]\to\mathbb{R}$ with $f(0)=0$ and derivative existing and bounded by 1 on $[0,1]\setminus X$, ...

**11**

votes

**1**answer

526 views

### How to prove that a monotone function is differentiable at some point?

This fact, which eventually belongs to Lebesgue, is usually proved with some measure theory (and we prove that the function is differentiable a.e.). Is there a significantly different approach? Let me ...

**7**

votes

**2**answers

118 views

### List of Bernoulli chaotic systems

Which discrete chaotic systems are known to be Bernoulli (i.e. measure theoretically isomorphic to a Bernoulli shift, one-sided or two-sided)?
I am aware that it is known for some uniformly ...

**0**

votes

**0**answers

77 views

### Hoeffding's lemma for unbounded r.v with bounded exponential map

Let $X$ be a real r.v with $E[e^{\lambda X}] < \infty $ for all $\lambda \in [-c,c]$.
Is it possible to get an Hoeffding's lemma like bound on $E[e^{\lambda(X-EX)}]$. That is, an upper bound: ...

**3**

votes

**0**answers

76 views

### Lower semi-continuity of the Hellinger-Fisher-Rao distance

I am currently working on unbalanced optimal transport, where the Hellinger (or sometimes Fisher-Rao) distance
$$
...

**2**

votes

**1**answer

140 views

### Classification of Lebesgue-Rokhlin spaces

I am currently trying to grasp some ideas on Lebesgue-Rokhlin spaces from Bogachev, "Measure Theory", vol. 2.
Such spaces are also known as standard probability spaces but the definitions are not ...

**0**

votes

**0**answers

50 views

### Volume growth of balls II

Let $b:(0,\infty)\to (0,\infty)$ be monotonically increasing.
Call $b$ limit-tight, if
$$
\lim_{\varepsilon\to 0}\ \limsup_{T\to\infty}\frac{b(T-\varepsilon)}{b(T)} =\lim_{\varepsilon\to 0}\ ...

**0**

votes

**1**answer

126 views

### Change of variable for integration with respect to Haar measure

I know how to estimate the integral* (see the update)
\begin{gather}
\int f(Ub)d\mu(U), \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ [2]
\end{gather}
where ...

**0**

votes

**0**answers

65 views

### Why is $\mathcal{E}(X)=\mathcal{E}(X,X^*)$?

According to a course about $\sigma$-agebras in infinite dimensional space they said that it is easy to see that :
$$\mathcal{E}(X)=\mathcal{E}(X,X^*)$$
where:
$X$ is separable real Banach space.
...

**0**

votes

**0**answers

157 views

### On a certain set of probability measures on a shift

Denote by $\mathbb{Z}_2=\{0,1\}$ the integers modulo 2.
Let $S:\mathbb{Z}_{2}^{\mathbb{N}}\times\mathbb{Z}_{2}^{\mathbb{N}} \rightarrow \mathbb{Z}_{2}^{\mathbb{N}}$ be the sum $S(a,b) = a+b$, where ...

**1**

vote

**1**answer

204 views

### What is the cofinality of the positive measure sets of reals?

What is the minimal cardinality of a family of sets of real numbers, each of positive Lebesgue measure, such that every set of real numbers of positive Lebesgue measure contains some member of the ...

**4**

votes

**1**answer

139 views

### Invariant subspaces are reducing subspaces in $L^2(\mu)$; where $\mu$ is a singular measure w.r.t Lebesgue measure

I have already posted this question on math.stackexchange but didn't get any answer. I hope this is the right place to ask this question.
Recently I was reading a book "Operator Function and system" ...

**2**

votes

**1**answer

79 views

### Compactness of cadlag martingales w.r.t. to the point-wise topology

Given a sequence of cadlag (right-continuous with left limits) martingales $X^n=(X^n_t)_{0\le t\le 1}$, we may use the well known criteria to determine whether it is weakly convergent, i.e. subtract a ...

**2**

votes

**1**answer

99 views

### Lower bounds from Fourier dimension?

According to Mattila, Geometry of sets and measures in Euclidean spaces, p. 168, the Fourier dimension $\text{dim}_F(A)$ of $A\subseteq \mathbb R^n$ is the unique number in $[0,n]$ such that for any ...

**1**

vote

**1**answer

139 views

### Averaging measurable functions over amenable group actions

Let $G$ be an amenable group acting on a space $X$.
Amenability means there is a $G$-invariant mean on $L^\infty(G,{\mathbf R})$.
Given a bounded function $f\colon X\to {\mathbf R}$ one can use the ...

**7**

votes

**2**answers

428 views

### A result of Sierpiński on non-atomic measures

There is a classical result commonly attributed to W. Sierpiński that reads as follows:
Theorem 1. If $f: \Sigma \to \bf R$ is a non-atomic (*) measure on a set $S$, then for every $X \in \Sigma$ ...

**0**

votes

**1**answer

112 views

### entropy growth of invariant measures - General question

In general, given a sequence of shift-invariant measures $\eta_{n}$ on $\{0,1\}^{\mathbb{N}}$ what to do to guarantee this convergence of entropies: $$h(\eta_{n}) \rightarrow \log2?$$
Because I'm ...

**0**

votes

**0**answers

52 views

### skorokhod integral as Weak Integral

Is it possible to express the skorokhod integral on a Banach space $B$ as a special case of the weak (or Pettis) integral over an appropriate Banach space $E$?
For example if $E$ is the space of ...

**1**

vote

**1**answer

110 views

### entropy and d-bar: how do we estimate continuity?

Let $G = \{0,1\}^{\mathbb{N}} = \mathbb{Z}_{2}^{\mathbb{N}}$ be the Bernoulli space of two symbols, let $\sigma$ be the shift map and $M(G)$ the set of $\sigma$-invariant probabilities. Let $\bar{d}$ ...

**1**

vote

**1**answer

85 views

### Equivalent measures on algebra also equivalent on $\sigma$-algebra?

Suppose $\mu$ and $\nu$ are finite positive measures on a measurable space $(X,\mathcal A)$. Let $\mathcal G$ be an algebra of $\mathcal A$. If $\mu$ and $\nu$ are equivalent on $\mathcal G$ in the ...

**0**

votes

**0**answers

97 views

### Optimization over space of probability measures

Consider an optimization problem as follows:
$$
\min\mathbb E_w[f_0(w)] \mathrm{\,\,\,\,\,\ s.t.\,\,\,\,} E_w[f_i(w)]\leq 0 ,\,\,\, i=1,\dots, k
$$
where the maximum is taken over $\mathscr M$, ...

**4**

votes

**2**answers

174 views

### Density of Gaussian measures on Banach spaces

I am trying to get my head around this question and was reading (1) which states the same a little bit more general:
Let $X$ be a separable Banach space and $X^*$ the dual space. The mean
value ...

**3**

votes

**0**answers

384 views

### “Nicely” strong measure zero sets

This question is essentially an expanded version of the unanswered half of Two strengthenings of "strong measure zero".
A set $X$ of reals is strong measure zero if, for any $f: ...