-1
votes
0answers
52 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 ...
2
votes
0answers
69 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
2answers
131 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, ...
2
votes
0answers
149 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
0answers
132 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
2answers
235 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
0answers
74 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
1answer
156 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
1answer
111 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
1answer
187 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
1answer
116 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
1answer
131 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 $\mathcal{R}$ ...
0
votes
0answers
89 views

Convergence of functions of bounded variation

Let $u_n$ be a sequence of smooth functions with $||u_n||_{W^{1,1}}<C<\infty$. By $BV$ compactness, a subsequence of $u_n$ converges to some $u$ in $L^{n/(n-1)}$. Assume $\int_{\Omega} (|\nabla ...
1
vote
0answers
67 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
3answers
191 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
3answers
276 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
1answer
155 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
1answer
133 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
1answer
185 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
3answers
427 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
1answer
208 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
1answer
119 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
0answers
79 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
0answers
114 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
0answers
120 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
1answer
132 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 ...
1
vote
1answer
145 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
1answer
196 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
1answer
187 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
1answer
255 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: ...
3
votes
0answers
215 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
0answers
145 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
0answers
240 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
1answer
117 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
1answer
175 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
0answers
164 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 ...
1
vote
0answers
137 views

The space $L_{1}(\beta\mathbb{N},\mu)$ [closed]

Is the space $L_{1}(\beta\mathbb{N},\mu)$ ($\mu$ is a regular Borel measure on $\beta\mathbb{N}$ and not purely atomic) isomorphic to $L_{1}[0,1]$ with the usual Lebesgue measure?
6
votes
2answers
114 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
0answers
164 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 ...
9
votes
0answers
302 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 ...
1
vote
1answer
178 views

Density of certain functions in $C_c^\infty(0,T;V)$ in the space $W(0,T) \approx H^1(0,T;V)$?

EDIT: I need to think more about the question I want to ask given comments in the answer below. Please close the thread if required. I leave it undeleted because answer is useful. Let $V \subset H ...
4
votes
1answer
214 views

Norms for complex measures

I'm searching for a state of the art introduction to norms on the space of complex measures (on $\mathbb R^n $, for example, or some compact subset thereof). I'd be interested in inequalities of the ...
3
votes
1answer
188 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 $E=L^{2}[a,b]$ ...
2
votes
2answers
544 views

The image of a measurable set under a measurable function.

Let $f:X \rightarrow (Y, \mathcal{Y})$ be an abstract function, with $\mathcal{Y}$ a $\sigma$-algebra on $Y$. Endow $X$ with $f^{-1}(\mathcal{Y})$. Is then $f(X)$ a measurable set in $Y$? If not, are ...
4
votes
4answers
484 views

Existence of dominating measure for weak*-compact set of measures

I have posted the following question also here a longer time ago, but due to no answers I thought it might fit better to MO. Let $(\Omega,\mathcal F)$ be a measurable space and $\mathcal P$ a ...
0
votes
1answer
184 views

Variation on Fatou's lemma for Sobolev norms

Recall that Fatou's Lemma says that for every sequence $f_n$ of non-negative measurable functions $$\int \liminf_{n\to \infty} f_n \ d\mu\leq \liminf_{n\to \infty} \int f_n\ d\mu \ .$$ If I am not ...
2
votes
1answer
366 views

When does a $W^*$-algebra have a standard Borel spectrum?

EDIT: André Henriques has commented below that the correct separability condition is not weak-* separability as I have written below, but separability of the predual. This post came out a bit long, ...
1
vote
0answers
116 views

Regular Borel Measures equivalent definition

Please help me understand how the below definition is equivalent to the standard definition of regularity which says that a measure is regular if for which every measurable set can be approximated ...
-1
votes
1answer
165 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 ...
4
votes
3answers
524 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 ...