**14**

votes

**0**answers

725 views

### A Kakeya-like problem: must a union of annuli fill the plane?

Let $S$ be a subset of $\mathbb{R}^2$ with the following property. For all $x \in \mathbb{R}^2$ and $\varepsilon \gt 0$, there exists a nontrivial interval $[a,b] \subseteq [1-\varepsilon,1]$, such ...

**13**

votes

**0**answers

312 views

### A question about small sets of reals

In ZFC, does there exist an uncountable set of reals $A$ such that for every closed measure zero set of reals $B$, $ A + B = \{a+b : a \in A, b \in B\} \neq \mathbb{R}$?
This question is motivated ...

**13**

votes

**0**answers

521 views

### Which complete Boolean algebras arise as the algebras of projections of commutative von Neumann algebras?

Projections in an arbitrary commutative von Neumann algebra form a complete Boolean algebra.
Moreover, a morphism of commutative von Neumann algebras induces
a continuous morphism of the corresponding ...

**11**

votes

**0**answers

421 views

### Does this metric have an official name? Lévy metric? Ky Fan metric?

Let $X$ and $Y$ be random variables taking values in a separable metric space $(S,d)$. The metric I have in mind is
$$\rho(X,Y) = \mathbb{E}[\min\{d(X,Y),1\}]$$
if $X$ and $Y$ take values in the a ...

**11**

votes

**0**answers

664 views

### surprisingly difficult filtration problem

I am interested in a proof of the following statement which seems intuitive, but is somehow really tricky:
Let $X$ be a stochastic process and let $(\mathcal{F}(t) : t \geq 0)$ be the filtration ...

**10**

votes

**0**answers

739 views

### Quotients of Measurable Spaces?

Let $(\Omega,\Sigma)$ be a measurable space and $\Pi$ be a partition of $\Omega$. There is a projection $\pi:\Omega\to\Pi$ that maps each $\omega\in\Omega$ to the unique partition cell in $\Pi$ ...

**9**

votes

**0**answers

136 views

### Haar measurable sets and quotient maps

Let $G$ be a locally compact Hausdorff group with a Haar measure $\mu$, let $H$ be a closed normal subgroup of $G$, and let $q: G \to G/H$ be the quotient homomorphism. Let $\nu$ be a Haar measure ...

**9**

votes

**0**answers

315 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 ...

**9**

votes

**0**answers

563 views

### Errata for the Treatise of Analysis of Dieudonné

I was looking again at the beautiful and quite complete work of Dieudonné, his Treatise of Analysis, to refresh my memory about some aspects of classical analysis. I especially love this Treatise for ...

**9**

votes

**0**answers

381 views

### Example of a quasi-Bernoulli measure which is not Gibbs?

Let $X=\{0,1\}^{\mathbb{N}}$. For simplicity I consider measures on $X$ only.
A measure $\mu$ is quasi-Bernoulli if there is a constant $C\ge 1$ such that for any finite sequences $i,j$,
$$
C^{-1} ...

**8**

votes

**0**answers

475 views

### Is every $\sigma$-algebra of sets *abstractly* the Borel algebra of a topology on perhps some other set?

Is every sigma-algebra the Borel algebra of a topology?
inspires the present question which asks for less.
Question: Given a $\sigma$-algebra ${\cal A}$ on a set $X$, does there exist a topology ...

**8**

votes

**0**answers

229 views

### Common extension of two sigma-additive measures

Let $\mathcal{A_1}$ and $\mathcal{A_2}$ be $\sigma$-algebras of subsets of some space X. Suppose $\mu_j$ is probabilistic measure on $\mathcal{A}_j$ for $j=1,2$. What are the necessary and sufficient ...

**8**

votes

**0**answers

489 views

### G-delta of measure 0 containig the rationals.

It is well known that $\mathbb{Q}$ is not a $G_\delta$. In fact no countable dense subset of $\mathbb{R}$ is a $G_\delta$. We order the rationals in a sequence: $\mathbb{Q}=\lbrace r_k\rbrace$, and ...

**7**

votes

**0**answers

145 views

### Qualitative weakenings of probabilistic independence

In probability theory, independence of random variables is characterised by
$$(1)~~X~\text{independent}~Y \; \iff \; P_{(X,Y)} = P_X \otimes P_Y \enspace ,$$
where $P_{(X,Y)}$ is the joint probability ...

**7**

votes

**0**answers

255 views

### Uniform closure of subspaces of Baire class 1

Describe a uniformly closed linear subspace $A \subset C([0,1])$ such that the space $B_1(A)$ is not uniformly complete. Here $B_1(A)$ is the set of all bounded functions $f$ which are pointwise ...

**7**

votes

**0**answers

369 views

### Is the set of real-valued lower semi-continuous functions measurable in epigraph topology (= topology of Gamma convergence)?

Let LSC = LSC([0,1]) be the set of non-negative, lower semi-continuous functions on the unit interval which take values in $\mathbb{R}_+ \cup \{\infty\}$. We use epigraph topology on LSC, i.e. a ...

**6**

votes

**0**answers

136 views

### Is an infinite-dimensional “Lebesgue measure” uniquely determined by a set of positive finite measure?

Let $\mu$ be a probability measure on a subset $C \subset \mathbb{R}^\infty$ of the space of sequences, and assume, for simplicity, that $C$ is closed and convex.
We say that $\mu$ admits shifts if ...

**6**

votes

**0**answers

128 views

### Cramer's theorem in Hilbert spaces

I am interested under what conditions Cramer's theorem applies in random variables taking values in Hilbert spaces. Following these lecture notes, but using a Hilbert space:
Let $X_1,X_2,\cdots$, be ...

**6**

votes

**0**answers

290 views

### Prokhorov's theorem for finite signed measures?

Prokhorov theorem provides a useful characterization of relatively compact sets w.r.t. narrow topology (topology induced by narrow convergence) in the space of probability measure.
Notation used ...

**6**

votes

**0**answers

244 views

### Why has Sacks' “Measure-theoretic uniformity” not been more influential?

In the 1969 paper "Measure-theoretic uniformity in recursion theory
and set theory," Trans. Amer. Math. Soc. 142 1969 381–420, Sacks gave
a measure-theoretic approach to several results previously ...

**6**

votes

**0**answers

180 views

### Generating stationary, ergodic random fields on a homogeneous space

Consider a homogeneous space $M$, which for the sake of concreteness, let's take to be $M = \mathbb R^d$. Fix some space $A$, and consider the space of functions $X = C(M,A)$, along with its Borel ...

**6**

votes

**0**answers

292 views

### “Liftings” of L^\infty functions

This is motivated by this question: Is there an inclusion of $L_\infty(G)$ into $C_0(G)^{**}$? and Bill Johnson's comments there.
Let $X$ be a locally compact Hausdorff space and $\mu$ a Radon ...

**6**

votes

**0**answers

478 views

### Density of countably additive measure in the set of all finitely additive measures.

Let $S$ be a countable discrete set, the following two results are quite easy to prove:
Every countably additive probability measure $\mu$ on $S$ commutes (in Fubini's sense) with every finitely ...

**6**

votes

**0**answers

400 views

### The log kernel and Bochner Theorem

I was wondering if it possible to find a measure $\eta$ on $\mathbb{R}$ such that
$$
L(x):=\log\frac{1}{|x|}=\int e^{itx}\;d\eta(t)
$$
for every $x\in [0,1/2]$.
On a structural ground, this ...

**5**

votes

**0**answers

185 views

### Approximating a measurable function from a second-countable, locally compact Hausdorff group to a separable Banach space

Let $ G $ be a second-countable, locally compact Hausdorff group and $ B $ a separable Banach space.
We say that a function $ f: G \to B $ is Bochner-measurable if and only if it is the everywhere ...

**5**

votes

**0**answers

174 views

### Finitely additive measures on $\mathbb Z_2^\omega$ with invariance and independence constraints

Let $G = \mathbb Z_2^\omega$, with pointwise addition. Assume the Axiom of Choice. I am interested in finitely additive probability measures $\mu$ defined on all of $\mathcal PG$ that can be ...

**5**

votes

**0**answers

282 views

### Skorohod theorem (weak convergence) on a discrete setting

I have a question about the application of Skorohod representation theorem. The questions arises in this paper about robust hedging in mathematical finance. It is about the very last equation on page ...

**5**

votes

**0**answers

211 views

### Existence of an universally measurable pullback

Let $X,Y$ and $Z$ be standard Borel spaces:
topological spaces homeomorphic to Borel subsets of complete separable metric spaces.
Let $K\subseteq X\times Y$ be analytic. Assume that $K_x$ is not ...

**5**

votes

**0**answers

104 views

### Simultaneous Strong Law of Large Number classes?

Say that $C$ is a SSLLN class of subsets of some topological space $V$ provided that for every sequence of i.i.d.r.v.s $X_1,X_2,...$ with values in $V$, we almost surely have: For every $A$ in $C$, ...

**5**

votes

**0**answers

184 views

### Diffusion processes in wide generality

It is common knowledge among schoolchildren that one may define jump diffusion processes in wide generality.
Hard question: What are the most general structures on which one may define something ...

**5**

votes

**0**answers

237 views

### Proof of Lomnicki and Ulam on Infinite Product Probability Spaces

Given an arbitrary, nonempty family $(\Omega_i,\Sigma_i,\mu_i)_{i\in I}$ of probability spaces, there exists a probability measure $\mu$ on $\otimes_i\Sigma_i$ such that for every finite set ...

**5**

votes

**0**answers

309 views

### Independent Events Inducing Probability Measures

Let $\mathcal{F}$ be a sigma algebra over $\Omega$ and $M$ the set of all probability measures on $\mathcal{F}$. Let $\mathcal{C}$ be some collection of pairs $(A,B)$ with $ \ A,B\in\mathcal{F}$. Now ...

**4**

votes

**0**answers

85 views

### Large Deviations: Exponential decay in normed spaces

Let $(X_1,X_2,\cdots)$ be a sequence of independent and identically distributed random variables taking values in some general normed space $(V,||\cdot||)$. Denote $\mu=E[X_1]$ and ...

**4**

votes

**0**answers

236 views

### Progressively measurable vs adapted

I often see in stochastic calculus books the terms 'adapted process' and 'progressively measurable process'. I know there is a small difference between them (every progressively measurable process is ...

**4**

votes

**0**answers

180 views

### Do all $L^{\infty}(\mu)$ spaces have the Grothendieck property?

Consider $L_{\infty}(\Omega,\Sigma,\mu)$, where $(\Omega,\Sigma,\mu)$ is any measure space. Does it it have the Grothendieck property? If the measure space is localizable, then it is true. The ...

**4**

votes

**0**answers

171 views

### Conditional expectation with respect to random closed sets

Short question
If $X$ is a random closed set, and $Y$ is an integrable random variable, I would like a definition of the "conditional expectation" $$\mathbf{E}[Y \mid X\ni x].$$ Has this been worked ...

**4**

votes

**0**answers

223 views

### decreasing rearrangements: why the asymmetry of measure-preserving maps?

Ryff proved in 1970 that the decreasing rearrangement $f^*$ of a, say, continuous function $f:[0,1]\to\mathbb{R}$ admits a measure preserving map $\phi$ such that $f=f^*\circ\phi$. In general it is ...

**4**

votes

**0**answers

65 views

### Importance sampling of finite path of stochastic difference equation

Before passing to question, let me briefly recap what's importance sampling of random variables is about. Suppose $\xi$ is a real-valued random variable with density $f$, and let $g:\Bbb R\to \Bbb R$ ...

**4**

votes

**0**answers

136 views

### Beck-Chevalley for measures?

A measurable set is a pair $(X,\Sigma)$ where $X$ is a set and $\Sigma$ is a $\sigma$-algebra on $X$. The elements $U\in\Sigma$ will be considered as subsets $U\subseteq X$. A morphism of measurable ...

**4**

votes

**0**answers

171 views

### Convergence of probability measures on a generating field of a sigma-field

Let $(\Omega,\mathcal{B})$ be a measurable space and let $\mathcal{F}$ be a generating field of $\mathcal{B}$. Assume $\mathcal{F}$ is standard, i.e. it is countable, and any normalized, non-negative, ...

**4**

votes

**0**answers

433 views

### What relates to measure spaces as topological spaces relate to metric spaces ?

Has there been study of a generalization of measure spaces along the following or similar lines ?
Given a measure space $(X, \Sigma, \mu)$, define for $U\in\Sigma$ a $\mu$-ball of radius $r$ of $U$ ...

**4**

votes

**0**answers

228 views

### Two standard probability spaces

Let $(X,\cal{A},\mu)$ be a standard (Lebesgue-Rokhlin) space with complete probabilistic measure (for example, $[0,1]$). Let $\cal{B}\supset \cal{A}$ be a wider then $\cal{A}$ $\sigma$-algebra on $X$, ...

**4**

votes

**0**answers

185 views

### The ring generated by measures

Suppose $X$ is a space equipped with a $\sigma$-algebra $\mathcal{M}_X$. Then the set of measures on $X$ is closed under addition and scalar multiplication by elements of ${\mathbb R}$. Formally ...

**4**

votes

**0**answers

133 views

### On the multidimensional Mellin transform of measures

Consider an integral transform of Borel measures supported on $\mathbb{R}^n_+$ given by
$$
f(z) =\int\limits_{\mathbb{R}^n_+} x^{z}\frac{\mu(dx)}{x}
$$
where $z = (z_1,...,z_n) \in \mathbb{C}^n$, ...

**4**

votes

**0**answers

370 views

### Measure Theoretic view of Hardy Littlewood Circle Method

Is it possible to view the Hardy-Littlewood Circle method as the Fourier transform with respect to the Lebesgue measure on [0,1) for an appropriate generating function defined in terms of additive ...

**4**

votes

**0**answers

709 views

### Exceptional Set in Egoroff's Theorem

I'll use the version of this question I posted on Stakexchange to replace the former version. To simplify the situaton, we assume the measure space we are dealing with right now is a closed interval ...

**4**

votes

**0**answers

335 views

### Conditional probabilities in Banach spaces

This is the infinite-dimensional sequel to my question, Conditional probabilities are measurable functions - when are they continuous?.
Let $\Omega = \Omega_1 \times \Omega_2$ be a probability space ...

**3**

votes

**0**answers

63 views

### How to define Product of Conditional Measures?

I have been wondering how to define the product of conditional measures as defined by Renyi-Popper. I spell the details below.
If $(X,\Sigma)$ is a measurable space, then the function $\mu : ...

**3**

votes

**0**answers

151 views

### Total variation and Hellinger distance inequality between truncated Gaussians

We know that the total variation distance, $d_{TV}(P,Q) = \frac{1}{2}\left|\left|P-Q\right|\right|_1$, between any two distributions $P$ and $Q$ is lower bounded by their squared Hellinger distance, ...

**3**

votes

**0**answers

77 views

### Pointwise a.e. formulation of parabolic PDE, what if null set depends on test function?

Let $u \in L^2(0,T;V)$ with $u_t \in L^2(0,T;V^*)$ be a solution of
$$\langle u_t(t), v \rangle + a(t;u(t), v) = \langle f(t), v \rangle$$
where $f \in L^2(0,T;V^*)$ and we have the usual assumptions ...