**3**

votes

**1**answer

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 $S\subset X$ ...

**1**

vote

**1**answer

93 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: ...

**7**

votes

**0**answers

134 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 ...

**4**

votes

**0**answers

92 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 ...

**2**

votes

**0**answers

92 views

### Regularity of measures in the theorem of Riesz

There are two concurrent theories of measure/integration on a locally compact topological spaces: either as positive linear forms on the space of continuous functions with compact support, or as Borel ...

**3**

votes

**0**answers

69 views

### Measurability for disintegration of a kernel

Let $(x, A) \mapsto P(x, A)$ be a probability kernel whose "target" (wikipedia terminology) is a product space $Y \times Z$, and say both $Y$ and $Z$ are compact metric spaces. For every $x$ there is ...

**0**

votes

**1**answer

85 views

### Functional representation of adapted jointly measurable stochastic processes

It seems like the question stated here in MSE has no answer yet and seems therefore for me to be not of a basic question type. For this reason I move it to MO.
Let $X_t : \Omega \to E, \ t \geq 0$ be ...

**3**

votes

**0**answers

77 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 : ...

**2**

votes

**2**answers

274 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 ...

**4**

votes

**1**answer

120 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 ...

**4**

votes

**0**answers

201 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, ...

**5**

votes

**0**answers

206 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

**1**answer

117 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 ...

**0**

votes

**0**answers

87 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 ...

**1**

vote

**0**answers

67 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 ...

**2**

votes

**1**answer

124 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 ...

**3**

votes

**0**answers

78 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 ...

**0**

votes

**0**answers

60 views

### Showing that a particular function from a locally compact Hausdorff group $ G $ to a $ C^{*} $-algebra $ A $ is Bochner-measurable

Suppose that we have the following data:
A $ C^{*} $-algebra $ A $.
A locally compact Hausdorff group $ G $.
A strongly Borel mapping $ \alpha: G \to \text{Aut}(A) $, the automorphism group of $ A ...

**1**

vote

**1**answer

178 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 ...

**6**

votes

**1**answer

183 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 \; ...

**13**

votes

**1**answer

1k views

### Is every continuous function measurable?

This question has already been asked on Math StackExchange here, but was too old to be migrated, and I think will be more appropriate to MathOverflow.
In non-Hausdorff topology it is standard to ...

**1**

vote

**0**answers

84 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 ...

**1**

vote

**1**answer

495 views

### For what nonnegative measures $\mu$ does $\mu*e^{-|\cdot|}\in L^{\infty}$?

I am trying to characterize all measures on $\mathbb{R}$ such that
$$
\sup_{x\in\mathbb{R}} \: (\mu*f)(x)<+\infty,
$$
where $f(x)$ is some specific integrable functions, such as $f(x)=e^{-|x|}$, ...

**3**

votes

**1**answer

196 views

### Unusual augmentation of a filtration

consider a probablity space $(\Omega,\mathcal{F}, \mathcal{P})$ and a filtration $(\mathcal{F}^0_t)$. In general $(\mathcal{F}^0_t)$ doesn't satisfy the usual conditions (it is not both complete at ...

**3**

votes

**1**answer

169 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 ...

**0**

votes

**0**answers

135 views

### Integration by parts for multidimensional Lebesgue-Stieltjes Integrals

I am concerned with the following problem:
I am wondering if there exists any sort of integration by parts formula for a multidimensional Lebesgue-Stieltjes integral. In my case the integral is given ...

**2**

votes

**2**answers

108 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 ...

**0**

votes

**0**answers

218 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 ...

**7**

votes

**3**answers

342 views

### Haar Measure on Locally Compact Semigroups

I'm reading on Haar measure and we know that every locally compact group admits a Haar measure, is the same true for semigroups? if not, is there a class of semigroups that admits a Haar measure?
...

**7**

votes

**1**answer

245 views

### A conjecture about the measure estimates of a trigonometric polynomial

Formulation of the Conjecture
Let $\Omega =(0,\pi)\times (0,2\pi)\subset\mathbb R^2$ and let $\psi:\Omega\to \mathbb{R}$ defined by $$\psi(x,t)=\sum_{k\in S \,j\in S'} \sin(kx)\left( ...

**1**

vote

**0**answers

201 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 ...

**4**

votes

**1**answer

297 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 ...

**20**

votes

**2**answers

988 views

### construction of nonmeasurable sets

I have a history question for which I've had trouble finding a good answer.
The common story about nonmeasurable sets is that Vitali showed that one existed using the Axiom of Choice, and Lebesgue et ...

**2**

votes

**1**answer

93 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
...

**0**

votes

**0**answers

100 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 ...

**3**

votes

**3**answers

234 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 ...

**1**

vote

**1**answer

176 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 ...

**0**

votes

**0**answers

143 views

### A not defined notion in Friedman's article about Generalized Fubini's Theorem

I intend to study Friedman's article, A Consistent Fubini-Tonelli Theorem for Nonmeasurable Functions (http://projecteuclid.org/download/pdf_1/euclid.ijm/1256047607). I think since I had a modern ...

**4**

votes

**1**answer

140 views

### On a.e. approximate differentiability of certain continuous real functions

I have the following question:
If $f:[0,1]\to \mathbb{R}$ is a bounded continuous function of $\sigma$-finite variation in sense 1, then is it true that $f$ is approximately differentiable a.e. on ...

**5**

votes

**2**answers

305 views

### non-Borel set which intersects every compact in a Borel set

I remember hearing some time ago that there is a locally compact Hausdorff space $X$ and a non-Borel subset $E$ which intersects every compact set in a Borel set. (This would contradict Lemma 13.9 of ...

**0**

votes

**1**answer

97 views

### Measurable functions lifted onto a space of point measures are measurable

I've been reading [1] and attempting to prove statements given without proof. In the paper the authors construct a measurable space of measures over a base space, and as an aside show an elegant way ...

**2**

votes

**0**answers

89 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 ...

**3**

votes

**2**answers

254 views

### Product of Lebesgue and counting measures

Let $\mathbb R$ be endowed with the standard Euclidean topology and let $\widetilde {\mathbb R}$ denote the line endowed with the discrete topology. Let $\mu$ and $\nu$ denote the Lebesgue and ...

**3**

votes

**2**answers

92 views

### Is this generating family of a measurable space of point measures a pi-system?

I'm learning some probability and measure theory and working my way through the first few paragraphs of [1]. My question is perhaps too basic for Math Overflow, but I hope it is welcome here.
Point ...

**2**

votes

**0**answers

94 views

### Uniformization/measurable selection theorems

Let $X,Y$ be measurable spaces and $F\subseteq X\times Y$. We say that $f:X\to Y$ is a uniformization map for $F$ if $(x,f(x))\in F$ for each $x\in \pi_X(F)$ where $\pi_X$ is the left projection map. ...

**2**

votes

**0**answers

50 views

### Stronger version of linearity for functions of measures

Let $X$ be a standard Borel space, and $P(X)$ be space of Borel probability measures on $X$. It is also a standard Borel space if endowed with the topology of weak convergence, so we can integrate ...

**4**

votes

**0**answers

417 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 ...

**2**

votes

**1**answer

204 views

### Inverse of a Borel surjection

Let $X$ and $Y$ be standard Borel spaces, and let $f:X\to Y$ be a surjective Borel map. Does there exist a Borel inverse of $f$, that is a Borel map $g:Y\to X$ such that $f\circ g = \mathrm{id}_Y$.
...

**3**

votes

**1**answer

246 views

### Quotients of standard Borel spaces

Let $X$ and $Y$ be standard Borel spaces: topological spaces homeomorphic to Borel subsets of complete metric spaces. Given a surjective Borel map $f:X\to Y$, we get an equivalence relation ...

**3**

votes

**1**answer

125 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 ...