All Questions
146 questions
12
votes
3
answers
870
views
Measure theory in nuclear spaces
Much of the literature on measure theory in linear spaces focuses on the case of normed linear spaces (e.g., the outstanding book by Vakhania, or its sequel). However, nuclear linear spaces "as far ...
7
votes
0
answers
549
views
Counter-example to the completeness of the Wasserstein metric
$\newcommand{\P}{\mathcal{P}}$
Let $(E,d)$ be a complete metric space, let $\P(E)$ be the set of all probability measures on $(E,\mathcal{B}(E))$. Let $W_d$ be the $1$-Wasserstein (Kantorovich) ...
- 931
13
votes
4
answers
5k
views
What is known about the Gaussian measure of the unit ball in a Hilbert Space?
Let $X$ be an infinite dimensional separable Hilbert Space with norm $||\cdot||$ and let $\mu$ be a Gaussian measure on $X$ such that $\mu(X) = 1$. What do we know about $\mu(B(0,1))$, where $B(0,1)$ ...
- 25.5k
6
votes
0
answers
388
views
Closedness of a set of measures, where conditional marginals are in closed $\varepsilon$-ball w.r.t. Wasserstein distance
Let $(E,d)$ be a bounded polish space (separable, complete metric space satisfying $\sup_{x,y\in E} d(x,y) < \infty$). By $\mathcal{P}(E)$ we denote the space of Borel probability measures on $E$ ...
- 8,071
4
votes
1
answer
1k
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 $f(x)=e^{-|x|}$, ...
CommunityBot
- 1
1
vote
1
answer
183
views
Diffuse measure space as a product of $[0;1]$ and another diffuse measure space
The title speaks of itself. How far is an arbitrary finite diffuse measure space from being almost isomorphic to a product of $[0;1]$ with another diffuse measure space? What would be reasonable ...
7
votes
0
answers
3k
views
What is vague convergence and what does it accomplish?
For convenience, let's say that I have a locally compact Hausdorff space $X$ and am concerned with probability measures on its Borel $\sigma$-algebra $\mathcal{B}(X)$. Natural vector spaces to ...
CommunityBot
- 1
1
vote
0
answers
96
views
Infimum of equivalent measures
Suppose I have a functional of the form
$$
F(\mathbb{P})\triangleq \int_{\mathbb{R}^d} \int_{\Omega}f(x,\omega)\mathbb{P}(d\omega)m(dx),
$$
where $m$ is the Lebesgue measure and $\mathbb{P}$ is a ...
- 173
9
votes
2
answers
548
views
What mode of convergence is this?
I'm interested in a new (to me) mode of convergence which is stronger than convergence in measure/probability. I want to know if it has a name and if it is used much in the literature. I will write ...
CommunityBot
- 1
2
votes
1
answer
115
views
Normalization of Gaussian w.r.t. Gaussian in a Banach space
I would like to compute
$$\int_X \exp\left(-\frac{1}{2}(Au)^2\right)\mathrm d\mu_0(u)$$
with a linear and continuous operator on a Banach space $A:X\to \mathbb R$ (in my case $X=C([0,1])$) and $\mu_0$ ...
- 34.7k
1
vote
1
answer
165
views
Decomposition of $L^2$-spaces and singular measures
If $\langle \Omega, \mathfrak{F}, \mathbb{P}\rangle$ is a measure space and $L^2$ is the corresponding $L^2$ space and
$$
K\oplus K^{\perp} \cong L^2(\mathfrak{F},\mathbb{P}).
$$
Then let:
$$
\...
- 29.8k
5
votes
0
answers
178
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 $\mathscr{P}$ be a probability ...
- 2,306
2
votes
1
answer
357
views
Measurability of integrals with respect to different measures
Let $Y$ be a locally compact Hausdorff topological space (further assumptions like metrizability, separability, etc., may be added if necessary) and let $\mathscr Y$ denote the Borel $\sigma$-algebra ...
CommunityBot
- 1
2
votes
0
answers
168
views
Interchanging integrals and continuous linear forms in RKHS
I am reading Reproducing kernel Hilbert spaces in probability and statistics by A Berlinet, C Thomas-Agnan.
In Chapter 5 INTEGRATION OF $\mathcal{H}$-VALUED RANDOM VARIABLES they write One of the ...
- 619
3
votes
1
answer
1k
views
If $H$ is a separable Hilbert space, is its dual dense in $L^2(H)$?
Let $H$ be an infinite-dimensional, separable Hilbert space, and let $\gamma$ be a Radon probability measure on $H$ with mean zero and covariance operator the identity $I$.
Let $H^*$ denote the space ...
CommunityBot
- 1
18
votes
4
answers
1k
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 ...
- 10.5k
8
votes
1
answer
360
views
Can we recover a topological space from the collection of Borel probability measures living on it?
Let $(X, \tau)$ be a topological space, and $\mathcal{P}(X, \tau)$ be the Borel probability measures living on $X$. Can we recover $(X, \tau)$ from $\mathcal{P}(X, \tau)$?
- 34.7k
1
vote
1
answer
720
views
Question about uniform continuity under Skorokhod Metric
Let $D=D([0,1], \mathbb{R})$ be the space of cadlag functions $x$ with $x(0)=0$ and $x$ is continuous on $1$. If we endow $D$ with Skorokhod Metric, see:
http://en.wikipedia.org/wiki/C%C3%A0dl%C3%A0g ...
- 1,872
-1
votes
1
answer
148
views
Continuity of function mapping $\mathcal{P}(\mathcal{P}(X))$ to $\mathcal{P}(X)$ [closed]
Given a topological space $Y$, let $\mathcal{P}(Y)$ be the set of all probability measures on $Y$, endowed with the weak* topology.
Let $X$ be a topological space (for convenience, it might be Polish ...
- 2,381
21
votes
2
answers
3k
views
A measure on the space of probability measures
This question was originaly posted in the stackexchange https://math.stackexchange.com/questions/1226701/a-measure-on-the-space-of-probability-measures but since it only got a comment I decided to ...
- 7,197
1
vote
0
answers
260
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 $\triangle\left(X,\...
- 11
7
votes
1
answer
444
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 ...
- 4,010
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
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
9
votes
3
answers
654
views
measure with given push-forwards
Let $X,Y$ be locally compact spaces (in my specific case, they are locally compact groups). Suppose that we are given a measure $\mu$ on $X$ and a finite number of quotient maps $p_1,\ldots,p_n:Y\...
CommunityBot
- 1
4
votes
1
answer
444
views
PDE-Based Triangle Inequality for Optimal Transportation
Suppose $\Omega$ is a suitably regular domain in $\mathbb{R}^n$ and $\rho_0,\rho_1\in\textrm{Prob}(\Omega)$. Benamou and Brenier showed that the $L_2$ transportation distance between $\rho_0$ and $\...
CommunityBot
- 1
4
votes
0
answers
309
views
Conditional expectation with respect to random closed sets
Short question
If $X$ is a random closed set, and $Y$ is an integrable random variable, I would like a definition of the "conditional expectation" $$\mathbf{E}[Y \mid X\ni x].$$ Has this been worked ...
- 6,287
4
votes
4
answers
3k
views
Convergence of probability measure and the *-weak convergence ?
Given a Polish space $X$, I note $C_b(X)$ the set of the continuous bounded functions with the norm of the uniform convergence, and $(C_b(X))^\star$ its topological dual with the $*-$weak convergence $...
- 166
6
votes
3
answers
1k
views
Is the set of the absolutely continuous functions a Borel set of the space of the continuous functions?
Does anyone knows whether the set of the absolutely continous functions $F :[0,1]\to \mathbb{R}^d$ of the form $$F(t)= a + \int_0^tf(s) ds$$ where $f$ is an integrable function is a Borel set of the ...
- 29.8k
-1
votes
1
answer
259
views
Absolute continuity of probabilities on Polish spaces and open sets. [closed]
On a polish space $\mathcal{X}$ i consider two Borel probabilities $P$ and $Q$ such that for any open set $E$ of $\mathcal{X}$ we have : $P(E) =0$ implies $Q(E)=0$. Does this imply that $Q$ is ...
CommunityBot
- 1
5
votes
1
answer
758
views
Cameron-Martin theorem for non-Gaussian measures
Let $X$ be a locally convex topological linear space, and $\mathbb P$ be a probability measure on $X$. Denote the mean vector $m \in X$ and covariance operator $k : X^* \to X$. Let $\tau_u : X \to X$ ...
- 29.8k
4
votes
1
answer
580
views
Density of linear functionals in $L^2$
Let $X$ be a locally convex topological linear space, and let $\mathbb P$ be a probability measure on $X$. Suppose that $\operatorname{var}(\varphi) < \infty$ for all continuous linear functionals $...
- 29.8k
2
votes
0
answers
136
views
equivalence of topologies defined on $M_1$(a subspace of bounded measures on $\mathbb{R}$)
Let $\mathcal{C}:=\mathcal{C}(\mathbb{R})$ be the space of continuous functions on $\mathbb{R}$ and $\mathcal{C}_b$ its subspace consisting of bounded elements. Define for $\phi(x):=1+|x|$,
$$
\...
- 3,958
2
votes
1
answer
245
views
Probability measures on $L^p$
Let $(X,\mathcal X,\mu)$ be a fixed measure space, and suppose that $\mu$ is stationary and ergodic with respect to the (left) action of a topological group $G$. Stationarity means that $\mu = g_* \mu ...
- 41
17
votes
5
answers
3k
views
Conditional probabilities are measurable functions - when are they continuous?
Let $\Omega$ be a Banach space; for the sake of this post, we will take $\Omega = {\mathbb R}^2$, but I am more interested in the infinite dimensional setting. Take $\mathcal F$ to be the Borel $\...
- 311
6
votes
1
answer
798
views
Prohorov's theorem for random elements of Hilbert space: weak convergence
Let $(\Omega,\mathcal{F},P)$ be a probability space and let $(E,\mathcal{E})$ be a separable Hilbert space ($E$) with Borel $\sigma$-algebra $\mathcal{E}$. For concreteness let us set $E=L^{2}[a,b]$ ...
- 23.2k
0
votes
1
answer
229
views
Weak convergence in measure for negligible sets.
Let $X$ be a Polish space and $(P_n)$ a sequence of Borel probabilities which converges weakly in measure to a Borel probability $P$. By this i mean that for any $f\in C_b(X)$ which is continuous and ...
- 17k
1
vote
1
answer
142
views
Linear Maps between $L^1$-spaces of singular measures
I posted the following question also here, but thought that I can get more answers in MO.
Let $(\Omega,\Sigma)$ be a measurable space and $\nu_1$, $\nu_2$ two probability measures on it. For $i=1,2$, ...
CommunityBot
- 1
2
votes
1
answer
469
views
If two probability distributions have the same weak limit and one of them satisfies Large Deviation Principle, what can we say about the other?
If the probability distribution function of two sequences of random variables have the same weak limit and one of the sequences satisfies a Large deviation principle, then does it imply that the other ...
- 586
4
votes
2
answers
427
views
Choice of predictable (or jointly measurable) eigenvalues and eigenvectors of nuclear-operator-valued stochastic process
Let $q^{ij}$, $i,j\in\mathbb{N}$, be predictable real-valued stochastic processes. Let $(e^i)$, $i\in\mathbb{N}$ be an ONB of a separable Hilbert space $H$. Assume that $Q=\sum_{i,j=1}^\infty q^{ij}...
CommunityBot
- 1
5
votes
0
answers
200
views
Diffusion processes in wide generality
It is common knowledge among schoolchildren that one may define jump diffusion processes in wide generality.
Hard question: What are the most general structures on which one may define something ...
CommunityBot
- 1
3
votes
2
answers
1k
views
Is there a corresponding Hahn decomposition theorem for the real-valued Radon measures?
Hello,
As we know that a signed measure $\mu$ on $R$ can be decomposed to the positive part $\mu_+$ and negative one $\mu_-$ by the Hahn decomposition theorem.
My question is whether each real-...
- 41.1k
-1
votes
1
answer
696
views
Can singular measures be viewed as vanishing distributions? (Answer No!)
Hello,
Here is my original question: let $\mu$ be a singular measure with respect to the Lebesgue's measure on $R$. Is it true that $\int \psi \mu(d x)=0$ for any test function $\psi\in C_c^\infty(R)$...
CommunityBot
- 1
9
votes
2
answers
674
views
Small crown probabilities (and infinite dimensional margin assumption)
My question is:
How do I find sharp upper bounds on $P(|q|\leq \epsilon)$ uniformly over a set of gaussian polynomes $q$ of degree two.
Notations and definitions (to make the question rigorous)
Let ...
- 96
3
votes
1
answer
1k
views
Cyl(E) = Borel(E) for E non-reflexive Grothendieck Banach space
This is sort of a follow-up to Borel(X) = \sigma(X') for X non-separable
PROBLEM: Given a Banach space $E$ over $\mathbb{K} \in \{\mathbb{C}, \mathbb{R}\}$ that has the Grothendieck property. ...
CommunityBot
- 1
7
votes
1
answer
1k
views
If $H$ is a separable Hilbert space, is $L^2(H)$ separable?
Let $H$ be a separable Hilbert space, and let $\gamma$ be a Radon probability measure on $H$ with mean zero and covariance operator the identity $I$.
Is the Hilbert space $L^2(H,\gamma)$ separable?
CommunityBot
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