Questions tagged [measure-theory]
Questions about abstract measure and Lebesgue integral theory. Also concerns such properties as measurability of maps and sets.
3,072 questions
9
votes
0
answers
624
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 ...
- 9,641
0
votes
0
answers
133
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 [t,...
- 19
2
votes
0
answers
448
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 ...
- 21
1
vote
0
answers
417
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 ...
- 213
0
votes
1
answer
529
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 $\...
- 28.2k
1
vote
0
answers
90
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 $\...
CommunityBot
- 1
9
votes
1
answer
992
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: ...
CommunityBot
- 1
2
votes
0
answers
98
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 ...
- 35
2
votes
1
answer
474
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 ...
- 3,968
1
vote
1
answer
174
views
Is it possible to define the density of the logistic map for $x<0$?
Probability density functions (PDF's) have inherent connections to the field of
Dynamical Systems.
The motivation for this question can be found in: http://www.stat.cmu.edu/~cshalizi/754/2006/notes/...
- 6,206
3
votes
0
answers
112
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}} |f(x_1,\dotsc,x_n)-f(x_1,\dotsc,x_{i-1},x'_i,x_{i+1},\dotsc,...
- 619
0
votes
0
answers
148
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 ...
- 424
3
votes
0
answers
250
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 ...
- 31
1
vote
1
answer
258
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 ...
- 1,616
3
votes
1
answer
179
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 $S\subset X$ ...
- 48.9k
5
votes
0
answers
137
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 $S_n=\frac{1}{n}[...
- 619
9
votes
0
answers
223
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 ...
- 619
0
votes
2
answers
435
views
conditional expectation under convex combinaison of probability measures(II)
Let $(\Omega,\mathcal{F})$ denote some measurable space. Let $P_1$ and $P_2$ denote respectively two probability measures. Now let $\mathcal{G}$ be some sub sigma-algebra of $\mathcal{F}$. Given a ...
- 26
6
votes
0
answers
365
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 ...
- 1,396
3
votes
0
answers
465
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 : \Sigma\...
- 191
3
votes
2
answers
470
views
If $ F(x,\bullet) \in {L^{2}}(G,B) $ for all $ x \in G $, then is $ x \mapsto F(x,\bullet) $ strongly measurable?
This question is related to something that I asked yesterday: If $ F(x,\bullet) \in {L^{\infty}}(G,B) $ for all $ x \in G $, then is $ x \mapsto F(x,\bullet) $ strongly measurable?
Pietro Majer ...
CommunityBot
- 1
2
votes
1
answer
104
views
Why do rotationally ordered configurations have well defined distributon function?
Let $u=(u_{j})_{j \in \mathbb{Z}}$ where $u_{j}\in \mathbb{R}$ for all $j \in \mathbb{Z}$ be a rotationally ordered configuration i.e. $S_{n,m}u>u$ or $S_{n,m}u<u$ or $S_{n,m}u=u$ where
$u<v$...
CommunityBot
- 1
7
votes
2
answers
2k
views
Simple functions on a product measure space
Let $ (X,\mathcal{F},\mu) $ and $ (G,\mathcal{G},\nu) $ be two measure spaces with $ \mu $ and $ \nu $ being $ \sigma $-finite. Per definition, the linear span of
$$
\{
\mathbf{1}_{C}
~|~
C \in \...
- 60.5k
5
votes
1
answer
298
views
If $ F(x,\bullet) \in {L^{\infty}}(G,B) $ for all $ x \in G $, then is $ x \mapsto F(x,\bullet) $ strongly measurable?
Let $ (X,\Sigma,\mu) $ be a $ \sigma $-finite measure space and $ B $ a Banach space. A function $ f: X \to B $ is said to be strongly $ \mu $-measurable iff it is the almost-everywhere pointwise ...
- 60.5k
4
votes
0
answers
1k
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, $...
- 41
3
votes
1
answer
156
views
Wiener measure of hitting sets A,B but not C (or easier hitting A but not C)
I am trying to formulate the measure of event
$E=\{B[0\infty)\cap A,B \neq \varnothing$ and $B[0\infty)\cap C= \varnothing\}$,
where $B[0\infty)$ is a Brownian path and $A,B,C$ are pairwise ...
CommunityBot
- 1
4
votes
1
answer
819
views
The Notion of Strong Measurability for Separable Banach Spaces
Let $ (X,\Sigma,\mu) $ be a measure space and $ B $ a Banach space. According to my understanding, a function $ f: X \to B $ is said to be strongly $ \mu $-measurable if and only if it is the almost-...
- 23.2k
0
votes
0
answers
454
views
Reference: Bochner Integral`
What would be an easily accessible book dealing with Bochner integration as applied to probability theory (I'm looking to understand random elements and their basic related concepts in a formal yet ...
- 5,405
16
votes
3
answers
1k
views
A natural center of a convex weakly compact set in Banach space
Question: Let $S$ be a convex weakly compact set in Banach space $H$. Propose a natural way to define the unique center $O \in S$.
Motivation: A lot! For example, in game theory $S$ can be a set of ...
- 423
4
votes
1
answer
2k
views
Lebesgue measure of boundary of Caccioppoli set
Can anything be said about the measure of the topological boundary of a Cacciopoli set in $R^n$? Of course, the reduced boundary has finite (n-1)-dimensional Hausdorff measure, but this does not say ...
CommunityBot
- 1
1
vote
0
answers
205
views
Inversion of Fourier transform of a multivariate gamma distribution in polar form?
Let $\mathbb{S}^{N-1}$ be the unit sphere in $\mathbb{R}^N$ under the Euclidean norm $||\cdot||$. Let $\mu$ be an infinitely divisible Borel measure. If there exists a finite measure $\alpha$ on $\...
- 11
2
votes
1
answer
440
views
Weak Convergence to Lebesgue Measure
I'm trying to understand the proof given by D. Rudolph in his paper "x2 and x3 invariant measures and entropy". I'm particularly trying to undestand the proof of lema 4.4.
Let's consider a secuence ...
- 23.2k
3
votes
0
answers
119
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 ...
CommunityBot
- 1
7
votes
1
answer
453
views
Do the terms of an iid sequence whose law has infinite expected value necessarily exceed the partial sums of the sequence infinitely often?
Let $\mu$ be a probability measure on $(0,\infty)$, and let $(\mathbf X_n)_1^\infty$ be a sequence of independent $\mu$-distributed random variables. Fix $\kappa > 0$, and consider
A) $\int x \; d\...
- 1,136
2
votes
1
answer
641
views
Fourier transforms of finitely additive bounded measures
Given a finitely additive positive regular bounded measure $\mu$ on ${\mathbb R}^n$ (i.e. a positive linear functional on $C_b({\mathbb R}^n)$), I wonder what can be said about its Fourier transform. ...
- 68
4
votes
1
answer
877
views
Kolmogorov doesn't show existence of Dirichlet process for arbitrary measurable spaces. Why?
I'm trying to understand the problem arising when using Kolmogorov's extension theorem to prove the existence of the Dirichlet process on an arbitrary measurable space $\left(\mathcal{X},\mathcal{A}\...
- 76
0
votes
0
answers
161
views
question about the tightness of probability measures for a general topological space
Let $(E,\mathcal{X})$ be a topological space and denote by $\mathcal{F}$ its collection of Borel subsets referred to $\mathcal{X}$. Now let $\mathcal{P}$ be the set of all probabilities on $(E,\...
- 1,835
3
votes
0
answers
910
views
Girsanov theorem with Geometric Brownian Motion
I am not a student in mathematics, but I am trying to use the following Theorem 8.6.6 (Girsanov theorem II) of Oksendal's SDE with geometric Brownian motion $S_{t}$ instead of the standard Brownian ...
- 24.8k
3
votes
1
answer
688
views
Is it possible to construct any random variable on the Euclidean Probability space?
Let $(\Omega,\mathscr A,P)$ be an arbitrary probability space,
and let $X:\Omega\to\mathbb R$ be a random variable.
Then,
one can generate a random variable $Y$ from the probability space $\big([0,1],\...
- 14.4k
2
votes
2
answers
139
views
Completion of the set of subsets with half volume.
Let $X$ be a measure space with finite $|X|=\int_X1$ and $f:X\rightarrow \mathbb{R}$ be a function. Under what condition on $X$ and $f$ does there exist a subset $Y \subset X$ satisfying the following?...
- 31.2k
0
votes
0
answers
232
views
Morphisms associated to measured spaces [duplicate]
In a previous discussion (von neumann algebras and measurable spaces), the connexion between von Neumann algebras and localized measured spaces was clarified. I would like to have a category theory ...
CommunityBot
- 1
4
votes
1
answer
2k
views
Uncountable family of random variables
Let $\{ \xi _a \}_{a \in [0;1]}$ be a family of independent uniformly distributed on $[0;1]$ random variables on some probability space
$(\Omega, \mathscr{F},P)$, indexed by a continuous parameter. ...
CommunityBot
- 1
6
votes
1
answer
1k
views
Proof of the Dunford-Pettis theorem
I would like to know where to find a complete proof of the Dunford-Pettis theorem:
A sequence $(f_n)_{n\geq 0} \subset L^1$ is uniformly integrable if and only if it is relatively compact for the weak ...
- 25.8k
2
votes
0
answers
162
views
Borel class of a set of measures
Let $K$ be a compact Hausdorff space and consider $X=Ball(M(K))$, the unit ball of the space of regular Borel measures on $K$. Endow $X$ with the weak-$*$ topology $\sigma(M(K),C(K))$, regarding $M(K)$...
- 1,704
1
vote
1
answer
367
views
How to get Fourier–Stieltjes transform on $\mathbb R$ from the nice function on $\mathbb T$ (periodic on $\mathbb R$)? [closed]
We put, $M(\mathbb R)= $The set of bounded complex Borel measure $\mu$ on $\mathbb R$ and for $\mu \in M(\mathbb R)$, we define $||\mu||:= |\mu| (\mathbb R) = \text {total variation of } \ \mu $; and
...
- 323
8
votes
3
answers
834
views
Do regular conditional distributions almost surely assign trivial measure to all members of the conditioning $\sigma$-algebra?
Let $(X,\Sigma)$ be a standard measurable space, let $\rho$ be a probability measure on $(X,\Sigma)$, and let $\mathcal{E}$ be a sub-$\sigma$-algebra of $\Sigma$. We will say that a stochastic kernel $...
CommunityBot
- 1
3
votes
2
answers
375
views
Infima of conditional densities after disintegration
Consider the measurable partition of the open unit square $(0,1)\times(0,1)$ into horizontal intervals $L_y=(0,1)\times\{y\}$. Let $\mu$ be a Borel probability measure with the disintegration
$$
\mu(...
- 708
0
votes
0
answers
184
views
Reference request: density of $C_c^{\infty}(\mathbb R^d)$ in $L^2(\mathbb R^d,d\rho)$
My question is motivated by an optimal transportation approach to PDE's and gradient flows in metric spaces (see e.g Otto's geometry of dissipative evolution equations: the porous media equation and ...
- 5,380
1
vote
1
answer
771
views
Bingham's paper "Finite additivity vs countable additivity" [closed]
I am reading N. H. Bingham's (2010) paper: ''Finite additivity vs. countable additivity''.
Page 8, he states:
"One area where the distinction between finite and countable additivity shows up most ...
- 2,758
3
votes
0
answers
69
views
Almost everywhere in a rectangle [duplicate]
I would like to ask a question about the product (Lebesgue) measure on rectangle. I tried to solve the problem but I couldn't.
Let $S$ be a subset of a region, say $R$ which is enclosed by a ...
