Questions about abstract measure and integral theory. Also concerns such properties as measurability of maps and sets.
5
votes
2answers
121 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
654 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
40 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
92 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
160 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
351 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
150 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
98 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
110 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
33 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
72 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
290 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
83 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
161 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
136 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
239 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
80 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
87 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
101 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
97 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
65 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
238 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
146 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
60 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
135 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: ...
3
votes
2answers
117 views
About a construction of Borel $\sigma$-algebra associated to a lattice
Let $(\mathcal{A}, \cup, \cap)$ a lattice (with minimum and maximum elements $\bot$ and $\top$).
Let $X\subset \mathcal{A}$ a generator set (a set of minimal cardinality that generate $\mathcal{A}$ ...
0
votes
1answer
172 views
Aronszajn measure
In a geometric measure theory (GMT) course I'm following this year, the professor told us about the Aronszajn measure, and asked us to go check by ourselves what it reprensents (the course was about ...
1
vote
1answer
173 views
Hoeffding's inequality for vector valued random variables
Is there a version of Hoeffding's inequality for vector valued random variables?
This seems to be hard to find and I wonder why. I suppose it is difficult to show Hoeffding's lemma, since the proof ...
3
votes
1answer
126 views
About the Caratheodory class.
Let $X$ a set, and $\mathcal{P}(X)$ the class of its subset's.
Let $\mathcal{A}\subset \mathcal{P}(X)$, we call a map $L: \mathcal{P}(X)\to[0, \infty]$ $\mathcal{A}$-regular if for any $S\subset X$ ...
