**1**

vote

**0**answers

25 views

### Stochastic calculus in $L^1$

Does there exist a more general (than Malliavin or Itô) "Stochastic calculus" defined on $L^1$ space, or some Orlicz space between $L^2$ and $L^1$?
For examples: are there:
Ito ...

**2**

votes

**0**answers

36 views

### Uniform convergence of long geodesic to Liouville measure

Here is the set up : let $(S,g)$ an hyperbolic surface and $L_g$ the associated volume measure. By the shadowing lemma there exist sequences of long closed geodesics, $\gamma_n$ which approximate the ...

**0**

votes

**0**answers

55 views

### Special random variables and monotone class theorem

I am currently reading a proof where the $\pi-\lambda$ Lemma and the monotone class theorem are applied to show a certain property for bounded random variables. The author of the book always shows the ...

**4**

votes

**3**answers

175 views

### Measure of intersections in probability spaces

Let $(X,\mu)$ be a probability space, and $0<\epsilon<1/2$. Let $\{A_i:i\in \mathbb{N}\}$ be a collection of measurable subsets of $X$ such that $\mu(A_i)\geq \epsilon$ for all $i\in\mathbb{N}$.
...

**2**

votes

**1**answer

65 views

### Covariance matrix as optimization problem solution?

I have seen the expectation of a random vector expressed as the solution to the optimization problem:
\begin{equation}
\mathbb{E}[X]=argmin_{v \in \mathbb{R}^n}\mathbb{E}[\|X-v\|_{l^2}^2](:= ...

**-1**

votes

**0**answers

27 views

### Total measure and Riesz theorem [migrated]

As I specified in the other question I asked, analysis is not my field, so I'm sorry if my question is trivial.
Riesz representation theorem states the following:
Given $X$ a locally compact ...

**15**

votes

**1**answer

302 views

### A nice subcategory of the category of measurable spaces

Is there some notion of "nice" measurable spaces and "nice" maps between them which satisfies the following properties?
The real line equipped with the Lebesgue $\sigma$-algebra is nice.
Any ...

**1**

vote

**0**answers

62 views

### Subgroups of finite non-zero Haar measure of abelian locally compact groups

Is it true that every subgroup of finite non-zero Haar measure of an abelian locally compact group should be open and compact? This is obviously true for the case of discrete abelian groups. Thanks.

**0**

votes

**0**answers

50 views

### Strong Lebesgue regularity lemma

I have troubles in understanding haw the Strong Lebesgue regularity lemma implies the Rademacher differentiation theorem. An informal proof was given by Tao on his blog: ...

**2**

votes

**1**answer

168 views

### What is the formal name of this set-related concept?

I "invented" a concept and it feels like it has already been invented before. I would like to know whether such a concept exists and if so, what is its name?
Let $S$ be a family of finite sets.
...

**1**

vote

**0**answers

73 views

### Can Gradient be controlled by Curl and Divergence in Morrey spaces

In $L^p(\mathbb{R}^3)$, it holds for $1< p< \infty$ and $\mu\in C^\infty_0(\mathbb{R}^3)$,
$$\|\nabla\mu\|_p\leq C \left( \|\operatorname{div} \mu\|_p + \|\nabla\times\mu\|_p \right).$$
So, how ...

**0**

votes

**0**answers

73 views

### weak-* versus entropy growth

General question. Let $\eta_{n}$ be a sequence of invariant measures on $\{0,1,2,...,p-1\}^{\mathbb{N}}$ and $B$ the Bernoulli uniform measure. Knowing that $\eta_{n} \rightarrow B$ in the weak-* ...

**-1**

votes

**0**answers

66 views

### Which functional can preserve the martingale property?

Let $M^n=(M^n_t)_{t\in [0,T]}$ be a sequence of continuous (or cadlag) martingales. Let $F : \mathcal D([0,T],\mathbb R)\to \mathbb R$ be some measurable function, where $\mathcal D([0,T],\mathbb R)$ ...

**9**

votes

**1**answer

311 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 ...

**3**

votes

**1**answer

112 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

**0**answers

105 views

### Conjecture about tail events

Let $(\Omega, \mathscr F, \mathbb P)$ be a probability space. Conjecture:
Suppose we have events $A_1, A_2, ...$ s.t. $\forall \ A \in \tau_{A} := \bigcap_n \sigma(A_n, A_{n+1}, ...)$, $P(A) = 0$ ...

**1**

vote

**1**answer

73 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

127 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

68 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

105 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),~ ...

**4**

votes

**1**answer

96 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

146 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

126 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

46 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

148 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

232 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

163 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

114 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

131 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

169 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

82 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 ...

**5**

votes

**0**answers

359 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

149 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 ...

**3**

votes

**1**answer

171 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

89 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 ...

**1**

vote

**0**answers

68 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

63 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

134 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

172 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

62 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

142 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

461 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

**1**answer

85 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

76 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

73 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

134 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

48 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

107 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

63 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.
...