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3 votes
2 answers
190 views

An uncountable measurable subset of $\Bbb R$ containing no nonempty perfect set

$\newcommand\R{\Bbb R}$Assuming the axiom of choice, is there an uncountable Lebesgue-measurable subset $S$ of $\R$ that contains no nonempty perfect set? Of course, such a set $S$, if it exists, ...
Iosif Pinelis's user avatar
11 votes
1 answer
500 views

Uncountable families of measurable sets with pairwise positive intersections

Let $(X,\mathcal{B},\mu)$ be an arbitrary finitely additive probability measure space, let $a>0$ and let $(A_i)_{i\in I}$ be an uncountable family of subsets with measure $\geq a$. Is there an ...
Saúl RM's user avatar
  • 10.6k
3 votes
0 answers
129 views

A Talagrand inequality for the supremum of partial sums over function classes under dependence. (Reference request)

As a consequence to the Talagrand concentration inequality, it is well known that for a measurable space $(S,\mathcal{S})$ and an i.i.d. sample $X_1,...,X_n$ of $S$-valued random variables, if $\...
Daan's user avatar
  • 141
3 votes
1 answer
109 views

Literature request: Covariance operators for Gaussian measures

I am looking to answer the question: If $\mathcal{B}$ is a separable Banach space and $R: \mathcal{B}^*\to\mathcal{B}$ is a symmetric and positive operator, then $\phi: \mathcal{B}^*\to\mathbb{R}, \...
ChocolateRain's user avatar
2 votes
0 answers
205 views

When should the empirical measure of an infinite sequence be defined?

Let $(x_n)_{n \in \mathbb{N}}$ be a (deterministic) sequence of nonnegative reals, possibly even with $x_n \in \mathbb{N}$ if you prefer. Then we'd like to define the empirical measure of such a ...
Tobias Fritz's user avatar
  • 6,406
1 vote
0 answers
87 views

Hausdorff distance and Hausdorff measure of symmetric difference

Let $X_n$ be a sequence of $k$-dimensional piecewise smooth submanifolds of $\mathbb{R}^m$, converging in Hausdorff distance to a $k$-dimensional piecewise smooth submanifold $Y \subset \mathbb{R}^m$, ...
Hajime S.'s user avatar
0 votes
0 answers
39 views

Comonotone solution for Optimal Transport problems with supermodular surplus

In Alfred Galichon's book Optimal Transport Methods in Economics the foollowing result is stated for OT problems on the real line. Theorem 4.3.(i) Assume that $\Phi$ is supermodular. Then the primal ...
Francesco Bilotta's user avatar
2 votes
1 answer
170 views

Law of large numbers for a continuum of Bernoullis

Suppose I have a family of $n$ independent Bernoulli random variables described by a vector of parameters $(p_i)_{i=1}^n$. As it is well known, the number of successes within this family is a random ...
Francesco Bilotta's user avatar
2 votes
1 answer
148 views

Is projection of a closed subspace Borel?

Specifically, letting $E$ be a separable infinite-dimensional real Banach space, and $D_2$ in $E\times E$ a closed linear subspace, is then $\{\,x:\exists\,y\,;(x,y)\in D_2\}$ a Borel set in $E\,$? ...
TaQ's user avatar
  • 3,584
2 votes
0 answers
101 views

Majorization theory on $\sigma$-finite measure spaces

I want to learn about majorization and submajorization theory on $\sigma$-finite measure spaces. I know things get a bit more complicated compared with the case of a finite measure spaces but I'm ...
Lau's user avatar
  • 759
11 votes
4 answers
950 views

Is there a name for finite unions of intervals?

Finite unions of intervals are simple sets that are used quite often, e.g. in measure theory. (The construction of the Cantor set is a noble example). I realised that I do not have a name for them. Is ...
Pietro Majer's user avatar
  • 60.5k
2 votes
1 answer
103 views

Sufficient conditions for the space of Radon measure to be a Banach space

Let $\mathcal{X}$ be a Hausdorff space and consider the space of Radon measures with bounded total variation $M(\mathcal{X})$ on $\mathcal{X}$. Usually, the additional assumptions on $\mathcal{X}$ are ...
ChocolateRain's user avatar
4 votes
1 answer
311 views

If $f=h\circ g$, then there's a measurable function $\tilde h$ such that $f=\tilde h\circ g$

Let $X,Y,Z$ be three standard measurable spaces and $f:X\to Z$ and $g:X\to Y$ two measurable functions. Suppose that there's a function $h:Y\to Z$ such that $f=h\circ g$. How can I show that there's a ...
rfloc's user avatar
  • 627
4 votes
1 answer
204 views

How probability-rich is the $\sigma$-algebra generated by a sequence of sets? (Sierpiński's theorem on non-atomic measures without using the AoC.)

$\newcommand\F{\mathcal F}\newcommand\si{\sigma}\newcommand\Om{\Omega}\newcommand\ep{\varepsilon}$Let $p\in(0,1)$ and let $(\Om,\F,P)$ be a probability space. Let $(A_n)$ be a sequence in $\F$ such ...
Iosif Pinelis's user avatar
2 votes
1 answer
385 views

De la Vallée Poussin criterion on uniform integrability for infinite measures

The de la Vallée Poussin criterion (which is often used in combination with the Dunford-Pettis theorem) is usually formulated for probability measures/finite measures, for example in [Bogachev: ...
kumquat's user avatar
  • 185
0 votes
2 answers
167 views

Is a signed measure $\mu$ on $\mathbb{R}^d$ characterized by the transform $\mathcal{L}_\mu (\lambda ):=\int e^{\langle \lambda,x\rangle }\mu (dx)$?

In the book "Probability Theory" by Achim Klenke there's the following theorem: a finite measure $\mu$ on $[0,\infty )$ is characterized by its Laplace transform $\mathcal{L}_\mu(\lambda):=\...
noob's user avatar
  • 15
0 votes
0 answers
54 views

Reference request: "doubly empirical" measure associated to a random measure

I am considering the following type of situation. Suppose we have a random probability measure, by which I mean a probability measure on a space of probability measures atop some Polish space $X$. In ...
pseudocydonia's user avatar
0 votes
1 answer
102 views

Lower bounds for truncated moments of Gaussian measures on Hilbert space

Let $\mu_C$ be a centered Gaussian probability Borel measure on a real separable Hilbert space $\mathcal{H}$ with covariance operator $C$. Denote the ball with radius $r$ in $\mathcal{H}$ centered at ...
S.Z.'s user avatar
  • 505
3 votes
0 answers
161 views

Lebesgue measure of the boundary of the positivity set of a function is zero?

Let $w$ be a function $\mathbb R^n\to \mathbb R$ with the following properties: $w$ is globally $\alpha$-Hölder continuous, $\alpha \in (0,1)$; $w$ is biharmonic on $C=\{w>0\}$; $w$ is subharmonic ...
Evelina Shamarova's user avatar
1 vote
1 answer
344 views

Is the Borel-Cantelli Lemma applicable here? [duplicate]

Consider $(X_{n})_{n\in\mathbb{N}}$ a sequence of random variables taking values in the set $\mathbb{Z}_{\geq 0}$ where $\mathbb{P}(X_{n} = i) > 0 $ for every $i\in\mathbb{Z}_{\geq0}$ which are ...
user1234's user avatar
  • 161
2 votes
0 answers
89 views

On dense subspaces of $L^p$-spaces of finitely additive measures

Let $\mu$ be a finite, finitely additive measure defined on the Borel $\sigma$-algebra of a real separable Hilbert space $\mathcal{H}$ with dual $\mathcal{H}^{*}$. Write $L^{p}(\mathcal{H},\mu)$ for ...
S.Z.'s user avatar
  • 505
1 vote
1 answer
307 views

Proof of the Dunford-Pettis theorem in the context of probability spaces

I'd like to know if there's a proof of the Dunford-Pettis theorem without using relatively advanced theorems of functional analysis such as Eberlein–Smulian Theorem. Since I'm only interested in ...
rfloc's user avatar
  • 627
1 vote
2 answers
262 views

Is the Boltzmann entropy lower semi-continuous in the weak topology induced by $C_b (\mathbb R^d)$?

For Lebesgue-absolutely continuous probability measures $\rho\ll \mathcal{L}^d$ in the whole space $\mathbb{R}^d$ with finite second moments (i-e $\rho\in \mathcal{P}^2_{ac}(\mathbb{R}^d)$), let $$ \...
Akira's user avatar
  • 835
4 votes
2 answers
255 views

Are the sublevel sets of Boltzmann entropy compact in Wasserstein distance?

For Lebesgue-absolutely continuous probability measures $\rho\ll \mathcal{L}^d$ in the whole space $\mathbb{R}^d$ with finite second moments (i-e $\rho\in \mathcal{P}^2_{ac}(\mathbb{R}^d)$), let $$ \...
Akira's user avatar
  • 835
3 votes
1 answer
220 views

Conditional expectation as square-loss minimizer over continuous functions

It is well-known that the conditional expectation of a square-integrable random variable $Y$ given another (real) random variable $X$ can be obtained by minimizing the mean square loss between $Y$ and ...
fsp-b's user avatar
  • 463
2 votes
1 answer
133 views

Can convergence in distribution necessarily be realised by almost-sure convergence?

Let $X$ be a Polish space. Let $(\mu_n)_{n \in \mathbb{N} \cup \{\infty\}}$ be a family of Borel probability measures $\mu_n$ on $X$ such that $\mu_n \to \mu_\infty$ weakly as $n \to \infty$. For each ...
Julian Newman's user avatar
1 vote
0 answers
168 views

Optimal transport-like problem where the objective depends on conditional probability distribution

$\DeclareMathOperator\marg{marg}$I would like to know if the following problem can be studied as an optimal transport problem, possibly imposing additional assumptions on the data. Consider two sets $\...
Francesco Bilotta's user avatar
0 votes
0 answers
161 views

Markov process with time varying transition kernels

I cross post this question from StackExchange as it may be more appropriate. I am interested in studying the evolution of a variable $\alpha_t\in [0,1]$ governed by the following stochastic dynamical ...
Francesco Bilotta's user avatar
1 vote
0 answers
87 views

Symmetry of the isoperimetric profile

Given a probability measure $\mu$ on a metric space $(X, \mathsf{d})$, the $(\mu-)$Minkowski content of a set $A$ is defined as $$\mu^+ (A) := {\lim\inf}_{r \to 0^+} \frac{\mu ( A_r \setminus A)}{r},$$...
πr8's user avatar
  • 801
3 votes
2 answers
102 views

Reference for Wiener type measure on $C(T)$ when $T$ is open

I'm considering Gaussian process on open domain $T$ in $\mathbb{R}^n$ and I tried to follow the abstract Wiener space construction of Gross. Since my sample paths are meant to be continuous, I thought ...
Kiyoon Eum's user avatar
1 vote
0 answers
97 views

Borel structure/sets coming from strong operator topology vs norm topology

Let $X, Y$ be Banach spaces. Moreover, let $\mathcal{L}(X,Y)$ be the space of bounded linear operators equipped with the standard operator norm topology, and $\mathcal{L}_{\mathrm{s}}(X,Y)$ the same ...
Marek Kryspin's user avatar
2 votes
0 answers
127 views

Measure algebra for a family of probability measures

Let $(X,B,P)$ be a probability space, $I_P$ the $\sigma$-ideal of $P$-null sets and \begin{align} B_P = B \ltimes I_P &= \{ A \mathbin{\triangle} N \mid A \in B, N \in I_P \} \end{align} the ...
Packo's user avatar
  • 285
3 votes
2 answers
273 views

Representing measurable map to compact space as a continuous map

Let $\Omega$ be a measurable space equipped with a $\sigma$-ideal $\mathcal{N}$ (though of as the "null sets"). Define the compact Hausdorff space $$ \tilde{\Omega} := \mathrm{Spec}(L^\infty(...
user avatar
0 votes
0 answers
134 views

Are there any books or literature on norms over measure space?

Consider the space of signed measures over some abstract space, we know the total variation norm makes the space Banach (I guess). So are some other norms. Are there some books or literature studying ...
Andrew Ying's user avatar
1 vote
1 answer
163 views

Convergence rate of a sequence of sets to a set-theoretic limit?

Suppose $n\in\mathbb{N}$ and set $A\subseteq\mathbb{R}^{n}$. If we define a sequence of sets $\left(F_r\right)_{r\in\mathbb{N}}$ with a set theoretic limit of $A$; how do we define the rate at which $\...
Arbuja's user avatar
  • 63
1 vote
1 answer
96 views

Asymptotic behavior of a Markov process on the set of $\{0,1\}$-polynomials

This question is cross-posted from https://math.stackexchange.com/questions/4711799/asymptotic-behavior-of-a-markov-process-on-the-set-of-0-1-polynomials I am trying to study the asymptotic behavior ...
Francesco Bilotta's user avatar
-1 votes
2 answers
407 views

Conditional expectation: commuting integration and supremum

Let $X$ and $A$ be compact Polish spaces endowed with Borel $\sigma$-algebras. Let $\mathcal{A} = X\times \mathcal{B}(A)$ be the $\sigma$-algebra consisting of cylinders whose projections on $A$ are ...
Vokram's user avatar
  • 109
2 votes
0 answers
130 views

Double quotient integral formula on $\Gamma \backslash G /K$

Let $G=\text{SL}(n,\mathbb R)$, $\Gamma=\text{SL}(n,\mathbb Z)$ and $K=\text{SO}(n,\mathbb R)$. Consider the double coset space $X= \Gamma \backslash G /K$ and its fundamental domain $\mathcal F\...
taylor's user avatar
  • 457
4 votes
2 answers
391 views

Lebesgue differentiation theorem at boundary points for Sobolev traces

$\newcommand{\R}{\mathbb R}$ Let $\Omega\subset \R^d$ be a smooth, bounded open set and fix $p\geq 1$. Fact 1: the usual Lebesgue differentiation theorem says that, if $u\in L^p(\Omega)$, then $$ u(x)...
leo monsaingeon's user avatar
7 votes
2 answers
180 views

Bisector of two points in a Riemannian manifold has measure $0$

Let $p,q\in M$, $p\neq q$, where $M$ is a Riemannian manifold. We will let the bisector of $p,q$ be $\mathcal{B}(p,q)=\{x\in M;d(p,x)=d(q,x)\}$. Does $\mathcal{B}(p,q)$ have measure $0$? I was ...
Saúl RM's user avatar
  • 10.6k
2 votes
1 answer
198 views

References on tilting distributions

I would be interested in any book, paper, or other reading material that gives a comprehensive treatment of tilted distributions using the following notion of "tilting" (or equivalent): ...
FD_bfa's user avatar
  • 147
8 votes
2 answers
960 views

Is there a measure theory for proper classes?

This question is naive, but I didn't get an answer at MSE: Is it straightforward to extend measure theory to proper classes? Of course when one tries to define measures on "large sets" ...
aduh's user avatar
  • 869
4 votes
1 answer
134 views

On partial absolute continuity

$\newcommand\B{\mathscr B}\newcommand\A{\mathscr A}\newcommand\si{\sigma}$Let $I:=[0,1]$, and let $\B$ and $\B^2$ denote the Borel $\si$-algebras over $I$ and $I^2$, respectively. Let $\A$ stand for ...
Iosif Pinelis's user avatar
1 vote
1 answer
262 views

Is $L_p(X, \mu, E)$ uniformly convex for $p \in (1, \infty)$ if $E$ is a uniformly convex Banach space?

Let $(X, \Sigma, \mu)$ be a $\sigma$-finite complete measure space, $(E, |\cdot|)$ a Banach space, and $p \in (1, \infty)$. Let $L_p := L_p(X, \mu, E)$ be the Bochner space of all $\mu$-integrable ...
Analyst's user avatar
  • 657
2 votes
1 answer
336 views

Complex Borel measures: does $\mu_n \to \mu$ weakly imply $|\mu|(\Theta) \le \liminf_n |\mu_n|(\Theta)$ for every open subset $\Theta$?

Let $\Omega$ be a metric space, $C_b(\Omega)$ the space of all real-valued bounded continuous functions on $\Omega$, and $\mathcal{M}(\Omega)$ the space of all finite signed Borel measures on $\Omega$...
Analyst's user avatar
  • 657
4 votes
2 answers
374 views

Vague convergence: confusion about the regularity of a signed Radon measure and that of its variation

I'm reading a proof of below theorem from this paper. Theorem A.3. Let $\Omega$ be a locally compact normal Hausdorff space. Let $\left\{\mu_n\right\} \cup\{\mu\} \subset \mathcal{M}(\Omega)$ and ...
Analyst's user avatar
  • 657
1 vote
1 answer
116 views

Strong subobject classifier for measurable spaces

Does the category of measurable spaces have a strong subobject classifier (specifically $2 = (\{0,1\}, \{\varnothing, \{0,1\}\})$? I would think the situation could be analogous to $\mathsf{Top}$, ...
Stefan Perko's user avatar
1 vote
0 answers
57 views

Uniform bound on the measure of $\Omega_\delta = \Omega \cap \delta\mathbb Z^d$ if $\Omega$ is an open bounded set with Lipschitz boundary

Let $\Omega \subset \mathbb R^d$ be an open bounded set with Lipschitz boundary. Let us consider $\Omega_\delta = \Omega \cap \delta\mathbb Z^d$ for $\delta >0$. I want to say that the measure of $\...
Hiro's user avatar
  • 131
1 vote
0 answers
74 views

"N-waves" (source-type solutions) for Hamilton-Jacobi equation $v_t + (v_x)^2 = 0$

Let us consider the Burgers equation $$u_t + (u^2)_x = 0$$ In Liu, Tai-Ping; Pierre, Michel, Source-solutions and asymptotic behavior in conservation laws, J. Differ. Equations 51, 419-441 (1984). ...
Riku's user avatar
  • 839
5 votes
0 answers
135 views

Criteria for tightness of Gaussian measures on Banach spaces

In Bogachev's book "Gaussian Measures" (Example 3.8.13) sufficient conditions for the (uniform) tightness of a sequence of centered Borel Gaussian probability measures on a separable Hilbert ...
S.Z.'s user avatar
  • 505

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