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

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10
votes
1answer
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
1answer
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
1answer
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
1answer
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
1answer
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
1answer
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
1answer
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
0answers
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
1answer
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
0answers
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
1answer
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
0answers
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
0answers
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
1answer
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
1answer
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
0answers
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
1answer
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
1answer
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
1answer
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
1answer
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
1answer
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
1answer
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
0answers
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
0answers
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
0answers
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
1answer
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
0answers
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
1answer
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
1answer
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
2answers
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
0answers
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
0answers
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
1answer
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
0answers
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
1answer
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
0answers
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
0answers
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
1answer
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
1answer
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
1answer
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
1answer
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
1answer
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
2answers
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
1answer
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
0answers
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
1answer
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
1answer
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
0answers
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
2answers
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
0answers
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: ...