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

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3
votes
0answers
26 views

Support of a Measure with Characteristic Functional Continuous in $L_p$, $1\leq p <2$?

Let $\mathcal{S}(\mathbb{R})$ be the space of smooth and rapidly decaying functions and $\mathcal{S}'(\mathbb{R})$ its dual, the space of tempered distributions. Let $\mu$ be a probability measure ...
4
votes
1answer
88 views

Function and its Gradient with Prescribed Norms

I'm not sure if the following question is too elementary for Mathoverflow. I'm sorry if it is the case. Question: Let $n\in\mathbb{N}$ and let $1\leqslant p<\infty$. Let $\alpha,\beta>0$. ...
0
votes
0answers
50 views

Volume of randomly changing sphere follows beta distribution

We are given $X,X_1,\ldots,X_N$ independent and identically distributed $k$-dimensional vectors. For a given query point $X_q\in\mathbb{R}^k$ assume without loss of generality that $X_1,\ldots,X_m$ ...
0
votes
0answers
32 views

Boundedness of a singular integral operator on $L^p(\mathbb{R})$, $1<p<\infty$

My singular integral operator is defined by \begin{align} Sf(x)=-\int_{-\infty}^{\infty}f(t-x) \frac{dt}{2\sinh\frac{\pi}{2}t}, \end{align} that is, a convolution $-\frac{1 }{2\sinh\frac{\pi}2x}\ast ...
5
votes
2answers
165 views

Rademacher average based Hoeffding Inequality

I am following these lecture notes: Given the i.i.d. $\mathcal{Z}$-valued random variables $Z_1,\dotsc,Z_m$ and $\mathcal{G}$ is a set of bounded functions $g\colon \mathcal{Z}\to[a,b]$. Corollary ...
30
votes
2answers
681 views

Translates of null sets

Does there exist a null set of reals $N$ such that every null set is covered by countably many translations of $N$?
0
votes
0answers
41 views

Taking power of the integrand in a Riemann-Stieltjie Integral

This is a problem I am trying to solve as part of a calculation for Value-at-Risk. Given that $P(X<x)=F(x)=\int_{\theta}F(x|\theta)dG(\theta)=1-\alpha$, where $F$ and $G$ are CDF's, is there a ...
1
vote
0answers
13 views

Outer measured induced by a point measure over an algebra of sets [migrated]

I'm helping a friend with problem 7 from section 3.3 of Vestrup's 'Theory of Measures and Integrals'. We got partway into the problem, but after a few hours of chatting about it we were still stumped. ...
4
votes
1answer
94 views

Does a surjective measurable map induce a surjective pushforward operator?

I hope it is OK to post a question that is basically the same as the months old currently unanswered question at math stackexchange Suppose X, Y are Polish spaces (without loss of generality, we may ...
6
votes
1answer
161 views

Connes' correspondences of two $L^\infty$-algebras

In his "Noncommutative Geometry" book Connes asserts (on p. 539) that for two standard probability spaces $(X,\mu_X)$, $(Y,\nu_Y)$ an $N$-$M$-bimodule for $M=L^\infty(X,\mu_X)$ and ...
15
votes
1answer
356 views

Integrals of pullbacks and the Inverse function theorem(s?)

The usual story goes like this: Smooth picture (?): For a smooth bijection $\phi: M \to N$ between $n$-manifolds the following is true: $\phi^{-1}$ is a local diffeomorphism a.e. ...
7
votes
1answer
155 views

Defining functions pointwise vs. almost everywhere (w.r.t. uncountably many mutually singular measures)

My question is motivated by a general measure-theoretic problem that one frequently encounters in probability: the need to work with uncountably many mutually singular measures at once, and with ...
2
votes
1answer
99 views

Is $ {C_{c}}(G) $ a meager subset of $ {L^{2}}(G) $ for a second-countable locally compact Hausdorff group $ G $?

The following problem is a stumbling block in a research project that I am working on: Problem. Let $ G $ be a second-countable locally compact Hausdorff group with a fixed Haar measure. Is it ...
1
vote
0answers
113 views

Generating the sigma algebras on the set of probability measures

I was wondering if somebody could help me see/provide a reference to the following fact: Let $X$ be a metrizable set, $\mathcal{F}$ the corresponding Borel sigma-algebra on $X$, and ...
1
vote
0answers
34 views

Progressive measurability and functional composition

Suppose we have a progressively measurable process $X$ taking values in $\mathbb{R}^d$. What are sufficient conditions on a function $f( x, t, \omega ) \colon \mathbb{R}^d \times [0,\infty) \times ...
1
vote
1answer
75 views

Conversion between condtional expection conditioned on $\sigma$-algebra and on r.v

Let $(\Omega, \mathcal F, P)$ be a probability space, and let $\mathcal G \subseteq \mathcal F$ be a sub-$\sigma$-algebra of $\mathcal F$ and $X : \Omega \to \mathbb R$ a random variable. Then the ...
5
votes
0answers
123 views

Characterizations of an exotic measure on the open sets in the circle $S^{1}$

Suppose that $U\subseteq S^{1}$ is open where $S^{1}=\{z\in\mathbb{Z}:|z|=1\}$. Then define $\mu_{n}(U)=\max_{t\in S^{1}}\frac{1}{n}\cdot|\{k\in\{1,...,n\}|t\cdot e^{\frac{2\pi ik}{n}}\in U\}|$. ...
0
votes
0answers
76 views

Discrete measures and discrete kernels

This is a cross-post from math.stack. Let $d\in\mathbb N$ and $\mu$ be the probability measure on $\mathbb R^d$ defined by $\mu=\sum_{k=1}^\infty 2^{-k}\delta_{x_k}$ for some sequence ...
3
votes
0answers
49 views

Bounds on truncated empirical mean

Let $(X_1,X_2,\cdots)$ be a sequence of independent and identically distributed random variables, $|X_i|\leq1$ and $S_n=\frac{1}{n}[X_1+\cdots+X_n]$. Is there an upper bound on the probability of ...
11
votes
1answer
292 views

Krein Milman theorem without the axiom of choice

The Krein-Milman theorem asserts that in a locally convex topological vector space, a nonvoid compact convex subset is the closed convex envelope of its extreme points. But I would like to know when ...
2
votes
0answers
80 views

Regularity of Dirac measure on Baire sets [closed]

Suppose $X$ is a locally compact Hausdorff space. Define the Baire sets in $X$, denoted by $\mathcal Ba(X)$, to be the smallest $\sigma$-algebra that contains all compact $G_\delta$ subsets of $X$. ...
0
votes
1answer
104 views

Is the push forward of a sigma-finite measure equivalent to a sigma-finite measure?

Let $X$ and $Y$ be locally compact, second countable spaces, and let $\varphi \colon X \to Y$ be a Borel map. Let $\mu$ be a sigma-finite measure on $X$. In general, the push-forward $\varphi_*\mu$ is ...
0
votes
0answers
56 views

Extension of functions on Cameron-Martin space

Edit: The following is more or less nonsense: Let $\mu$ be the Standard Gaussian measure on $\mathbb{R}^\infty$ (i.e. the measure such that the projections $p_j$ are independent $\mathcal{N}(0, ...
1
vote
0answers
56 views

Getting a measure from a premeasure through an adjoint

Let's take the category of measure spaces with objects $(X,\mathcal{F},\mu)$ and avoid the morphisms for now (I'm not sure what they should be), where $X$ is a set, $\mathcal{F}$ is a ...
1
vote
0answers
107 views

Is it possible to improve the order of convergence of averages of random variables if they are not identically distributed?

Let $X_n$ be a sequence of independent random variables (but not necessarily identically distributed) taking values in $[-1,1]$ that have the following property: 1) The average $A_n := \frac{(X_1+ ...
5
votes
1answer
269 views

Itô's article “A measure-theoretic approach to Malliavin calculus”

Apart from citations all over the internet, the following paper appears to be off-the-grid. K. Itô, A measure-theoretic approach to Malliavin calculus, in 'New Trends in Stochastic Analysis', Proc. ...
4
votes
1answer
84 views

Calculate Hausdorff measure with Frostman measures

Fix a metrix space $(X,d)$ and consider the Hausdorff (outer) measures $\mathcal{H}^s$ on $X$. A Frostman measure on $X$ is a finite Borel measure $\mu$ such that there exists $C,t,r_0>0$ with ...
1
vote
1answer
154 views

A question regarding sets of Vitali's type in models of $ZF+GCH$ where $L$$\neq$$V$

Consider sets of Vitali's type in models of $ZF+GCH$ where $L$ $\neq$$V$. Are there sets of Vitali's type in both $L$ and $V-L$? If so, is there any way one can distinguish the constructible sets of ...
4
votes
1answer
209 views

Characterizing residually amenable groups

Let $G$ be a finitely generated group. The amenability of $G$ is equivalent to the existence of a certain "weak measure" on $G$. Is there such a characterization for residually amenable groups as ...
1
vote
0answers
51 views

Measure algebras and their subalgebras

Let $A,B$ be two Boolean algebra with measures $m,p$ thereon, respectively such that the measure algebra $(A,m)$ is isomorphic to the measure algebra $(B,p)$. Suppose that we have two isomorphic ...
4
votes
0answers
163 views

Dual or pre-dual of BV

Was there any relevant work to determine the dual (or more likely the predual) of the space of bounded variation functions $BV(\mathbb{R}^n)$ (I recall the definition : a function in ...
12
votes
4answers
911 views

Sierpinski's construction of a non-measurable set

In the early 20th century there was a lot of fuss over the axiom of choice implying that there are Lebesgue non-measurable sets of reals. In his book about The Axiom of Choice, Gregory Moore points to ...
2
votes
2answers
140 views

Uniformly bounded operator family and pointwise convergence

Let $1 \leq p < \infty$ be fixed and let $\Omega \subseteq \mathbb{R}^n$ be open. Let $(Q_n)_{n \in \mathbb{N}}$ be a uniformly bounded family of operators on $L^p(\Omega)$, i.e. there exists ...
0
votes
0answers
56 views

Question about the representation of Skorokhod

I have a question about Skorokhod's representation theroem. Let $\Omega=R^m$ and define the canonical process $X=(X_1, ..., X_m)$, i.e. $X(\omega):=\omega$ for any $\omega=(\omega_1,..., \omega_m)\in ...
3
votes
0answers
65 views

Reasoning about dependent and independent quantities by “degrees of freedom”

In his classic textbook Foundations of the Theory of Probability Kolmogorov defines Independence a little bit differenent then it is usually done today. He denotes a probability space by $(E, \mathcal ...
0
votes
1answer
69 views

Extend product sigma-algebra to cross-constant sets

We have two probability spaces $(\Omega_1,\mathcal{F_1},P_1)$ and $(\Omega_2,\mathcal{F_2},P_2)$. Is it possible to construct probability space $(\Omega=\Omega_1\times\Omega_2,\mathcal{F},P)$ such ...
7
votes
0answers
241 views

Two questions about universally measurable sets

I have two questions about universally measurable sets: (1) Is there a universally measurable set of reals which does not have the Baire property? (2) Is there a universally measurable set of reals ...
0
votes
0answers
81 views

What is the sigma field of the derivative of a process?

When $t\to X_t$ is an absolutely continuous process ($X_t= X_0+ \int_0^t Y_s dt$ for some measurable process $Y_t$) we have for all $t$ $$\sigma(Y_t) \subset \cap_{\epsilon >0}\sigma(X_{s}, s\in ...
0
votes
0answers
88 views

Defining density of a random function using Radon-Nikodym Theorem

Let $(\Omega,\mathbb{F},P)$ be a probability space and $E$ be an infinite dimensional Banach space and $\mathbb{B}$ be the $\sigma$-algebra of Borel subset of $E$. Let $X$ be random function defined ...
2
votes
0answers
102 views

Lebesgue point and regularity of functions

A known theorem says that for $f \in L_{loc}^1(\mathbb{R}^d)$, almost every point is a Lebesgue point. I know too a theorem saying that for $f \in W_{loc}^{1,p}(\mathbb{R}^d)$ , every point is a ...
0
votes
1answer
98 views

Are the natural numbers a disjoint union of infinite sets of zero asymptotic density? [closed]

Suppose $\mathbb{N}=\bigsqcup_{i\in\mathbb{N}}E_i$ with $\#E_i=\infty$ for each $i$. Is it possible that $\limsup_{N\to\infty}\frac{1}{N}\#(E_i\cap\{1,\ldots,N\})=0$ for all $i$, which would mean ...
10
votes
3answers
319 views

Reference for a strong intermediate value theorem for measures

Let $\mu$ be a finite nonatomic measure on a measurable space $(X,\Sigma)$, and for simplicity assume that $\mu(X) = 1$. There is a well-known "intermediate value theorem" of Sierpiński that states ...
1
vote
0answers
66 views

Measurability of solution of diffusion equation in sub sigma algebra

I want to solve the following problem: Get $\omega \in \Omega \subset \mathbb{R}$, $x \in D \subset \mathbb{R}^2$ and $0<a_i\leq a(.,.)\leq a_x<\infty$. Let $a( x;. )$ and $f(x;.)$ be ...
6
votes
1answer
239 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 ...
1
vote
1answer
149 views

How does Azuma's Inequality result from Pinelis Inequality?

According to [1] Let $(\mathcal{X},||\cdot||)$ be a separable Banach space and let $S(\mathcal{X})$ denote the class of all sequences $f=(f_j)=(f_0,f_1,...)$ of Bochner-integrable random ...
1
vote
0answers
46 views

Finding a general form of the density function when we have a four dimensional random variable

Consider a subject having time of the specific event $T_i$, which is a single sample from a distribution $F_i$ with density $f_i$ and support $[t_{\min},t_{\max}]$, for $i= 1,\ldots,n$. Let these ...
2
votes
0answers
76 views

McDiarmid's inequality on normed spaces

McDiarmid's inequality says if a function $f: \mathcal{X}^n\to\mathbb{R}$ has the property that $$ \sup_{x_1,\dotsc,x_n,x'_i\in\mathcal{X}} ...
0
votes
0answers
63 views

existence of locally translation-invariant Borel measure on Frechet manifolds

It is well known that the only locally finite, translation-invariant Borel measure on an infinite-dimensional, separable Frechet space is the trivial measure. I am wondering about an analogous ...
3
votes
0answers
129 views

Hausdorff densities

I've been stuck on this one for a while now. Given an $\mathcal{H}^{s}$ measurable subset $E\subset \mathbb{R}^n$ with $0<\mathcal{H}^{s}(E)<\infty$, we let $\overline{D}^{s}(E,x)$ denote the ...
7
votes
1answer
142 views

Space of Borel measurable maps

That's a question from MSE (here) that did not receive any answer for some days. I migrate it to MO. Let $X$ and $Y$ be two standard Borel spaces and consider the set $M(X,Y)$ of measurable maps $f: ...