# Tagged Questions

**4**

votes

**1**answer

166 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 ...

**2**

votes

**1**answer

68 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

79 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 ...

**4**

votes

**0**answers

203 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 ...

**1**

vote

**2**answers

142 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 ...

**3**

votes

**1**answer

147 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 ...

**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 ...

**0**

votes

**1**answer

171 views

### A very natural question in weak* topology [closed]

Can you provide me a counter example for this.
Suppose that I have a sequence of probability measures
$(\mu_{r,t})_{r,t>0}$ on a compact space metric $X.$
Suppose additionally that:
there exists ...

**3**

votes

**1**answer

170 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 ...

**10**

votes

**1**answer

242 views

### smooth Luzin theorem

For measurable functions $f(x)$, $g(x)$ on $[0,1]$ define the distance $\rho(f,g)$ as a Lebesgue measure of the set $\{x:f(x)\ne g(x)\}$. Then Luzin's famous theorem states that $C[0,1]$ is dense with ...

**10**

votes

**1**answer

261 views

### Is this property equivalent to Lusin's property (N) for continuous functions?

A function $F:[0,1]\rightarrow\mathbb{R}$ satisfies Lusin's (N) property if for every measure zero set $A\subseteq [0,1]$, $F(A)$ has measure zero. (This includes the assertion that $F(A)$ is ...

**0**

votes

**1**answer

133 views

### Dense subsets on set space

Let $X$ be a metric space, and $\mathscr{B}$ the $\sigma$-algebra generated by open sets of $X$. Can we find a countable dense subsets of the metric space $(\mathscr{B},d)$ with the metric ...

**0**

votes

**0**answers

67 views

### Measure generated by Semigroup $\exp[-t|p|]$

I am studying Ingrid Daubechies' paper 'An Uncertainty Principle for Fermions with Generalized
Kinetic Energy'.
On page 514, the measure $\mu_{x,y;t}$ is introduced as generated by the semigroup ...

**3**

votes

**0**answers

103 views

### $f, \hat{f} \in L^{p}\cap L^{\infty} \implies f\in B(\mathbb R)$ (algebra of Fourier- Stieltjes transforms )?

For a bounded complex Borel measure $\mu$ on $\mathbb R$, we define, its Fourier-Stieltjes transform, $\hat{\mu}(y)= \int_{\mathbb R} e^{-2\pi ix\cdot y} d\mu(x); (y\in \mathbb R).$
Let $1\leq p \leq ...

**4**

votes

**2**answers

173 views

### Tightness of Measures, Riesz Representation for locally compact spaces

Let $X$ be a locally compact, separable metric space and $\mu_n$ a sequence of probability measures on $S$. Let $\mathfrak{C}$ be a convergence determining class for the weak topology (for instance, ...

**3**

votes

**0**answers

165 views

### Do all $L^{\infty}(\mu)$ spaces have the Grothendieck property?

Consider $L_{\infty}(\Omega,\Sigma,\mu)$, where $(\Omega,\Sigma,\mu)$ is any measure space. Does it it have the Grothendieck property? If the measure space is localizable, then it is true. The ...

**4**

votes

**0**answers

156 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 ...

**3**

votes

**2**answers

330 views

### Riesz's representation theorem for non-locally compact spaces

Every version of Riesz's representation theorem (the one expressing linear functionals as integrals) that I have found so far assumes that the underlying topological space is locally-compact. (For ...

**1**

vote

**0**answers

79 views

### Finding density of Haar measure relative to the Liouville volume measure on $\frac{G^{\mathbb{C}}}{P}$

Let $G$ be a connected compact Lie group and $G^{\mathbb{C}}$ be the complexification of the Lie group $G$ then we know that by polar decomposition we can write $G^{\mathbb{C}}\cong G\times ...

**3**

votes

**1**answer

193 views

### When is the bound in Riesz-Thorin Interpolation Theorem attained?

Let me recall the statement of Riesz-Thorin theorem (see also http://en.wikipedia.org/wiki/Riesz%E2%80%93Thorin_theorem).
Theorem (Riesz-Thorin): Let $(X,\mu)$ and $(Y,\nu)$ be $\sigma$-finite ...

**2**

votes

**1**answer

124 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 $\tau_u : X \to X$ ...

**3**

votes

**1**answer

206 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 ...

**0**

votes

**1**answer

128 views

### Some convergence similar to weak-$\ast$ convergence on the space of finite measures

I have a question:
Let $D$ be the space of cadlag functions defined on $[0,1]$ and $V$ be its subspace consisting of $x$ with finite variation and $x(0)=0$.
Define $TV(x)$ as the total variation ...

**0**

votes

**1**answer

156 views

### The dual space of the Dirac measures on an Abelian group

Let $G$ be a Hausdorff locally-compact Abelian group and $L^2(G)$ the Hilbert space of two-integrable complex functions on the group.
Question. What would be natural vector space $\mathcal{R}$ ...

**1**

vote

**0**answers

75 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 $\phi(x):=1+|x|$,
$$
...

**0**

votes

**3**answers

218 views

### dual space of a subspace of the space of bounded measures

Let $\mathcal{M}=\mathcal{M}(\mathbb{R})$ be the space of bounded measures. Equipped with the weak convergence, the dual space of $\mathcal{M}$ is $\mathcal{C}_b(\mathbb{R})$ consisting of continuous ...

**7**

votes

**3**answers

343 views

### Which Sigma-Ideals in a Sigma-Algebra are Ideals of Null Sets?

My question is motivated, to be somewhat vague, by an attempt to see how much a measure space is defined by the set of null sets. In other words, assume we are not given a concrete measure on a space ...

**1**

vote

**1**answer

175 views

### Fubini for distributions which are not measures?

We have a "nonnegative" distribution $\mu$ with compact support in $\mathbb{R}^2$ which is not a measure, as we can produce a linear function $f(x,y)=x-1$ such that the integral of $f^{2k}$ w.r.t. ...

**3**

votes

**1**answer

145 views

### On Radon measures with values in Banach space

It is known that continuous linear functionals on the space $C_0({\mathbb{R}^n})$ are bounded Radon measures ${\cal M}({\mathbb{R}^n})$ where $C_0({\mathbb{R}^n})$ is uniform closure of the space of ...

**2**

votes

**1**answer

194 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 ...

**9**

votes

**3**answers

567 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 ...

**7**

votes

**2**answers

303 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 ...

**-2**

votes

**1**answer

120 views

### a measure convolution equation

My question is:
Given a function $f$ in the Schwartz class, we are looking for a measure $\mu$ which is a solution of the convolution equation: $f = e^{-|.|^2/2} \ast \mu$, where $e^{-|.|^2/2}$ is ...

**1**

vote

**0**answers

86 views

### Differentiability of $f*g$ on the circle, for integrable f, bounded g, and some decay of the Fourier coefficients of f

If $f\in L^1(\mathbb{T})$ and $g\in L^\infty(\mathbb{T})$ where $\mathbb{T}$ is the circle, such that $\hat{f}\in L^{p}(\mathbb{Z})$ for some $1\leq p<\infty$, do we have that $f*g$ is ...

**0**

votes

**0**answers

124 views

### Dual of the space of vector valued Borel measures

What is the dual of the space of all vector valued Borel measures?

**1**

vote

**0**answers

124 views

### A problem concerning measures on locally compact spaces

I am stuck on a question for quite sometime now, although in the text it is said to be "apparent". The problem goes as the following :
Let $X$ and $Y$ be locally compact Hausdorff spaces. Then $M(X)$ ...

**1**

vote

**1**answer

148 views

### Lebesgue's integrability condition in several variables

The well known Lebesgue's condition of Riemann integrability says that a bounded function in one variable
$f\colon [a,b] \to \mathbb{R}$ is Riemann integrable if and only if it is continuous almost ...

**2**

votes

**1**answer

155 views

### functions of bounded variation and gradient vector measure

I want to prove a function of bounded variation on some domain $D\subset R^n$, $f\in BV(D)$, has the property that there is a constant $C$, such that
$$
\lim_{r\rightarrow 0}\frac{C}{r^{n+1}} ...

**2**

votes

**1**answer

206 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. ...

**2**

votes

**1**answer

199 views

### strong convergence sufficient conditions

Would it be true that $\mu_n \to \mu$ strongly if $\int f\mathrm{d}\mu_{n}\to \int f\mathrm{d}\mu$ for every uniformly continuous function? Assume the space is $\mathbb{R}^{N}$ and has the usual ...

**0**

votes

**1**answer

305 views

### What are some characterizations of the strong and total variation convergence topologies on measures?

I asked this question on StackExchange a few days ago but didn't get any response, so I thought I would try here.
The Wikipedia article on convergence of measures defines three kinds of convergence: ...

**2**

votes

**0**answers

221 views

### Dual of $L^2(0,T,C)$ where $C'=BD(\Omega)$

Here is the hypothesis of my problem : $T>0$, $\Omega$ is a bounded open subset of $\mathbb{R}^n$ with a regular boundary. I'm actually looking for a proof that explains what is the dual of ...

**2**

votes

**0**answers

156 views

### An inequality for Lp-functions

I am interested in the following inequality:
\begin{equation}
\int\limits_\Omega \left\vert f_1 - f_2 \right\vert^p ~ d\mu \le C_p \left[ \int\limits_\Omega \left\vert f_1 \right\vert^p ~ d\mu + ...

**6**

votes

**0**answers

271 views

### Prokhorov's theorem for finite signed measures?

Prokhorov theorem provides a useful characterization of relatively compact sets w.r.t. narrow topology (topology induced by narrow convergence) in the space of probability measure.
Notation used ...

**3**

votes

**1**answer

126 views

### Weak continuity of Lebesgue decomposition

Let $X$ be a space with its $\sigma$-algebra $\mathcal{B}$; we are given a finite measure $\mu$ and a sequence of finite measures $\nu_n$ such that, for every bounded continuous function ...

**4**

votes

**1**answer

182 views

### Doubt on Morrey spaces of measures according to T. Giga and Y. Miyakawa

I'm reading 'Navier-Stokes flow in $\mathbb{R}^3$ with measures as initial vorticity and Morrey spaces' (a paper of Giga and Miyakawa) but there is something that I don't understand about the ...

**2**

votes

**0**answers

179 views

### Cameron Martin space

I have seen two definitions of Cameron Martin space of a Gaussian measure $\nu$ on a Banach space (say $W$) and am unable to establish their equivalence. Any help would be appreciated.
1) It is the ...

**6**

votes

**2**answers

117 views

### Vanishing of integral on hemispheres implies vanishing of function?

Consider a function $F$ on the half space $\{(x,y,z)|z>0\}$. If $F$ is analytic, it is straightforward to show that
A) The integral of $F$ over the hemisphere $(x-x_0)^2 + (y-y_0)^2 + z^2 = R^2$ ...

**4**

votes

**0**answers

199 views

### decreasing rearrangements: why the asymmetry of measure-preserving maps?

Ryff proved in 1970 that the decreasing rearrangement $f^*$ of a, say, continuous function $f:[0,1]\to\mathbb{R}$ admits a measure preserving map $\phi$ such that $f=f^*\circ\phi$. In general it is ...

**8**

votes

**0**answers

343 views

### Does this metric have an official name? Lévy metric? Ky Fan metric?

Let $X$ and $Y$ be random variables taking values in a separable metric space $(S,d)$. The metric I have in mind is
$$\rho(X,Y) = \mathbb{E}[\min\{d(X,Y),1\}]$$
if $X$ and $Y$ take values in the a ...