**1**

vote

**0**answers

48 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 ...

**0**

votes

**0**answers

66 views

### Ask for a good reference for the calculus involving singular continuous measure [migrated]

I am not an expert on measure theory. I am sorry if this question is too simple for some experts here.
Suppose the measure $\mu$ is singular continuous on $\mathbb{R}$, such as the cantor measure. ...

**0**

votes

**1**answer

295 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

163 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

128 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

41 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

83 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

215 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 ...

**1**

vote

**0**answers

27 views

### Exterior measure using open balls [migrated]

I'm trying to show that given $A\subseteq\mathbb{R^n}$ the exterior measure of Lebesgue, $\mu^*$, can also be defined as
$$\mu^{*,B}(A) = \inf{\sum_{j}\mu(B_J)},$$
where the infimum is taken over the ...

**6**

votes

**3**answers

283 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

196 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

123 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

196 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

743 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

73 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

86 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

203 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

150 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

137 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

116 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

274 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

82 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 ...

**3**

votes

**0**answers

71 views

### Non Borel Spaces: Gauge Integral

Question
Is there a generalization of the gauge integral to measure spaces that do not necessarily arise out of some topology?
I'm wondering since it seems as the gauge crucially uses ...

**2**

votes

**0**answers

77 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 ...

**2**

votes

**2**answers

148 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

86 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

63 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

49 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

107 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

182 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

196 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

102 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 ...

**4**

votes

**1**answer

108 views

### Analytic enlargement of an analytic set

Let $X,Y$ be Borel spaces and $A\subseteq X\times Y$ be an analytic set. Let $\pi:X\times Y \to X$ denote the projection map onto $X$. Does there always exist a set $B$ such that $\pi(B) = X\setminus ...

**3**

votes

**0**answers

122 views

### Embedding probability spaces in the completion of $[0,1]^K$

Question: Can every probability space $(X,\scr F,\mu)$ be $\sigma$-embedded in the completion of the space $[0,1]^K$ (equipped with a product of Lebesgue measure) for some set $K$?
Here, $f:\scr F\to ...

**4**

votes

**0**answers

205 views

### Why does it seem that $rca=rba$? [closed]

The following paradox has got me stumped. I'm hoping someone can point out the error.
Take a locally compact metric space $X$ and define the $C_b(X)$ and $C_0(X)$ as the spaces of continuous ...

**3**

votes

**0**answers

64 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 ...

**1**

vote

**2**answers

144 views

### Measures, orthogonal to holomorphic functions

Let $G$ be a domain in $\mathbb{C}^{d}$ and let $H\left(G\right)$ be the space of all holomorphic functions on $G$.
My question is how to characterize all such (Radon) measures $\mu$ on $G$, that ...

**1**

vote

**2**answers

98 views

### Volume of normal cone of a simplex (at a vertex)

This question is related to an orthant-type simplex in $\mathbb{R}^n$, which can be defined as
$$ S = \{x\in\mathbb{R}_{+}^n: \sum_{i=1}^nx_i \leq 1\}=\overline{\mbox{conv}}(0,e_1,\ldots,e_n). $$
For ...

**1**

vote

**0**answers

96 views

### Make this marginalization statement rigorous

Intuition tells me that
$$ p(x\,|\,y) = \int p(x,\theta\,|\,y) \; d\theta$$
by the "law of marginalization", pretty much for any object $\theta$.
I would like to make this statement rigorous, ...

**3**

votes

**1**answer

151 views

### Extension of a bilinear functional

Does any one know an example of a bilinear functional $B:C(X)\times C(Y)\to {\bf R}$ ($X$ and $Y$ are open subsets of Euclid spaces) which cannot be extended continuously to a measure $\mu:C(X\times ...

**3**

votes

**1**answer

207 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. ...

**5**

votes

**2**answers

184 views

### Relationship of Euler product to coprimality densities for arbitrary sets of primes

Continuing the curiosity of my last couple questions: Is it the case that for every set of primes $F$, the asymptotic density of the integers coprime to all of $F$ is $\displaystyle \prod_{p \in F} (1 ...

**1**

vote

**1**answer

46 views

### Additivity of asymptotic density of periodic sets

It is well known that the asymptotic density of an infinite union of disjoint sets of integers may not be the sum of their individual asymptotic densities.
Can this failure of countable additivity ...

**3**

votes

**1**answer

115 views

### For Every Measure Zero Set $E$ There Exists a Positive Measure with Lower Lebesgue Density 0 and Upper Lebesgue Density 1

This is related to a question asked on mathstackexchange http://math.stackexchange.com/questions/831184/for-every-null-set-e-there-is-a-measurable-set-f-with-different-upper-and-lo. This question is ...

**1**

vote

**2**answers

218 views

### Question on separability of a measure

Following this question here this question come to mind.
Consider a measured σ-algebra $(S,\mu)$ . Assume that μ is normalized to have total weight 1, and that S is complete (contains all subsets of ...

**1**

vote

**1**answer

48 views

### Integral representation of joint projection valued measures

Given two positive $\sigma$-finite measures $\mu_{1/2}$ on the spaces $X_{1/2}$ one can define the product measure $\mu_1\otimes\mu_2$ on the product space $X_1\times X_2$. It can be proved that the ...

**0**

votes

**0**answers

49 views

### Approximate of binary function by indicator functions of one variable

Let $(X,F,u)$ and $(Y,G,v)$ be two probability spaces.
I know a result that the indicator functions of product measurable sets can be approximated by indicator functions of one variable. That is, for ...

**2**

votes

**0**answers

53 views

### Justification of the convolution operation of $L^1(G)$ functions where $G$ is a LCA group (measurability)

Suppose $G$ is a locally compact abelian Hausdorff group (LCA), and $\lambda$ is the Haar measure on it. We all know the convolution of two $L^1(\lambda)$ functions $f$ and $g$ on $G$ is defined as
...

**6**

votes

**2**answers

103 views

### How does one prove that $L_1(\mu)$ is weakly sequentially complete for any measure?

It is a theorem of Steinhaus that for any finite measure $\mu$, the Banach space $L_1(\mu)$ is weakly sequentially complete. Using the Radon-Nikodym theorem one can extend this easily to ...

**1**

vote

**1**answer

115 views

### Laplacian on space of measures

Let $X$ be a compact Riemannian manifold and let $\mathcal{M}(X)$ be the space of regular finite Borel measures with the total variation as norm.
The Laplace-Belrami-Operator $\Delta$ on $X$ with ...