Skip to main content

All Questions

Filter by
Sorted by
Tagged with
0 votes
0 answers
63 views

The union of weighted compact supported continuous function

Let $\Omega\subset \mathbb R^N$ be open. Given a weight function $v\geq 1$ such that $v\in L^1_{\text{loc}}(\Omega)$ and $l.s.c$. Also supposethere exists a Lipschitz continuous sequence $v_n$ such ...
JumpJump's user avatar
  • 679
3 votes
0 answers
229 views

Area defined with $\pm$ closedness

Denote $B_n\subset\Bbb R^n$ to be unit ball at origin. Denote $S\subset B_n$ to region of type $\mathsf I$ if it satisfies $$s\in S\iff\forall t\in S, s+t\in S\mbox{ or }s-t\in S$$ I am convinced $\...
Turbo's user avatar
  • 13.9k
4 votes
1 answer
393 views

How can I show that "almost all function" have property P?

The following is cross-posted from https://math.stackexchange.com/questions/1391293/is-almost-all-function-a-well-defined-concept since I didn't (yet) get an answer there. (I hope that's okay?) ...
mimuller's user avatar
  • 151
6 votes
1 answer
223 views

Is the space of vectorial functions that are Dunford integrable complete?

Let $X$ a Banach Space and $(\Omega, \Sigma, \mu)$ a measure space. A function $F:\Omega\rightarrow X$ is Dunford integrable if $x^\ast\circ F$ is $\mu$-integrable for every $x^\ast\in X^\ast$. The ...
Husaí Vázquez's user avatar
1 vote
1 answer
210 views

Is it true that for dimension d=3, if $v\in H_0^1(\Omega)$ but $v\notin L_\infty(\Omega)$ then exponent of v or -v is not summable?

I have the following Question: 1) Is it true that if $\Omega\subset\mathbb R^3$, $\Omega$ - bounded, $v\in H_0^1(\Omega)$ but $v\notin L_\infty(\Omega)$ implies $\int\limits_{\Omega}{e^vdx}=+\infty$ ...
Svetoslav's user avatar
  • 261
8 votes
2 answers
784 views

Is taking the product of signed measures weakly continuous?

For a Polish space $X$, let $C_b(X)$ denote the real Banach space of bounded continuous real-valued functions on $X$. Let $M(X)$ denote the space of all finite signed Borel measures on $X$, equipped ...
Nate Eldredge's user avatar
0 votes
0 answers
405 views

What does the Plancherel theorem say about positive-definite distributions?

I'm trying to understand the answer to this MO question: Bochner's theorem for measures of positive type, which suggests a relationship between Bochner's theorem and the Plancherel theorem. The ...
Tian An's user avatar
  • 3,799
4 votes
2 answers
558 views

Is a specific sequentially closed subset of $M([0,1])$ closed?

Let $M([0,1])$ be the set of finite signed measures on $[0,1]$ (with the topology generated by the sets $\left\{ \mu \in M([0,1]) : \left| \int f(x) \mu(dx)- a\right| \leq \delta\right\}$ for all $\...
Ori's user avatar
  • 95
5 votes
2 answers
1k views

Is the space of signed finite measures on a compact set $M([0,1])$ a sequential space?

Let $M([0,1])$ be the set of finite signed measures on $[0,1]$ (with the topology generated by the sets $\left\{ \mu \in M([0,1]) : \left| \int f(x) \mu(dx)- a\right| \leq \delta\right\}$ for all $\...
Ori's user avatar
  • 95
8 votes
1 answer
635 views

Is the Jordan decomposition of a self-adjoint functional constructive?

Let $A$ be an abstract C*-algebra, and let $\varphi\colon A \rightarrow \mathbb C$ be a bounded linear function. Assuming the axiom of choice, there exist unique positive bounded linear functions $\...
Andre Kornell's user avatar
3 votes
1 answer
355 views

convex function with distributional Hessian $D^2 f \leqslant \lambda$, $\lambda$-concave?

Let $f:R^n \to R$ be convex (may not $C^1$), $$[D^2f]=[D^2f]_{ac}+[D^2f]_s=[h_{ij}] L^n+[D^2f]_s$$ is the Lebesgue decomposition of the Hessian matrix. Where $[h_{ij}]$ is the density w.r.t the ...
mafan's user avatar
  • 471
3 votes
2 answers
968 views

Can one estimate the distribution of eigenvalues of a matrix by its Cauchy/Stieltje transform?

Given a real symmetric $n$ dimensional matrix $A$, with eigenvalues $\lambda_i$ I am defining its Cauchy transform as the function, $f_A(z) = \sum_i \frac{1}{z-\lambda_i}\,$ Is there any information ...
Student's user avatar
  • 617
4 votes
1 answer
209 views

Simultaneous approximation of arbitrary functions in Hölder space and in $L^2(\mu)$ by a smooth function and its derivative

Let $\mu$ be a probability measure on the circle $S^1=\mathbb{R}/\mathbb{Z}$ which is singular with respect to the Lebesgue measure $\lambda$. Consider the functions spaces $L^2(\mu)$ on the one hand, ...
Benoît Kloeckner's user avatar
7 votes
1 answer
306 views

An indicator of a planar subset as an element of a tensor product

Denote $I=(0, 1)$, and let $\mu$ be the Lebesgue measure on $I$. Does there exist a function $f$ on $I\times I$ viewed as an element of the space $L^\infty(\mu\times\mu)$ such that $$ f^2=f $$ (that ...
limanac's user avatar
  • 452
4 votes
2 answers
479 views

When does a function space allow for point evaluations? [closed]

Consider a space of (generalized) functions $F$ defined on a measure space $\Omega$ and equipped with a topology. What are necessary and sufficient conditions for point evaluations at arbitrary $x \...
shuhalo's user avatar
  • 5,327
-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 ...
user66910's user avatar
3 votes
3 answers
370 views

How to show that there's a continuous function separating convex sets of Radon measures?

First, the setup: $X$ is a compact set. By Riesz's representation theorem $C(X)^*=${all Radon measures on $X$}. $K$ is a convex, closed set of probability measures. $m$ is a probability measure out of ...
JustSomeGuy's user avatar
4 votes
2 answers
378 views

Topologies for which $\mathcal{M}(X)\otimes \mathcal{M}(Y)$ is dense in $\mathcal{M}(X\times Y)$

Are there complete TVS topologies for which $\mathcal{M}(X)\otimes \mathcal{M}(Y)$ is dense in $\mathcal{M}(X\times Y)$ This question is strongly linked to is the space of all borel measures on $\...
Duchamp Gérard H. E.'s user avatar
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 ...
Bruce Wayne's user avatar
6 votes
2 answers
1k views

Existence of a measure-preserving bijection

Let $f, g \, \colon \mathbb{R}^n \rightarrow \mathbb{R}$ be two Borel-measurable functions such that $f$ is non negative and g is radially symmetric, the function $ (0, \infty )\ni t \mapsto g (tx)$ ...
Mike's user avatar
  • 161
3 votes
3 answers
666 views

Is the space of all borel measures on $\mathbb R^n$ isomorphic to the tensor product of spaces of borel measures on $\mathbb R$?

Similarly to the decomposition $L_2(\mathbb R^n) = L_2(\mathbb{R})^{\otimes n}$ as vector spaces (and even as Hilbert spaces) , do we have $bm(\mathbb{R^n}) = bm(\mathbb{R})^{\otimes n}$ where $bm(\...
Elesthor's user avatar
4 votes
3 answers
472 views

Measure with `somewhere dense' support

Let $X$ be a compact Hausdorff (but not necessarily metrizable) space. Is it always true that there exists a probability Borel measure $\mu$ and an open set $U$ such that any nonempty open set $V\...
Joel Moreira's user avatar
  • 1,701
3 votes
0 answers
144 views

Transitive closure of balanced bounded mass transport

Given two $\sigma$-finite measures $\mu$ and $\nu$ on $\mathbb{R}^n$, write $\mu \sim \nu$ iff there exist countable decompositions $\mu = \mu_1 + \mu_2 + \cdots$ and $\nu = \nu_1 + \nu_2 + \cdots$ ...
James Propp's user avatar
  • 19.7k
3 votes
2 answers
675 views

distributional Hessian for semiconvex functions on non-smooth manifolds

Let $f:R^n \to R$ be convex. Then there exist signed Radon measures $\mu^{ij}=\mu^{ji}$ such that $$ \int_{R^n} f \frac{\partial^2 \varphi}{\partial x_i \partial x_j} dx= \int_{R^n} \varphi d\mu^{ij} ...
mafan's user avatar
  • 471
0 votes
1 answer
517 views

Injective inclusion map from RKHS function space to $L_p(\mu)$

Let $X$ be a measurable space, $\mu$ be a $\sigma$-finite measure on $X$, and $H$ be a separable reproducing kernel Hilbert space over $X$ with a measurable kernel $k$. At a certain part in a proof I ...
Eric's user avatar
  • 103
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 ...
Goulifet's user avatar
  • 2,306
3 votes
1 answer
145 views

Is $ {C_{c}}(G) $ a meager subset of $ {L^{2}}(G) $ for a second-countable locally compact Hausdorff group $ G $?

The following problem is a stumbling block in a research project that I am working on: Problem. Let $ G $ be a second-countable locally compact Hausdorff group with a fixed Haar measure. Is it true ...
Transcendental's user avatar
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,\...
Mark's user avatar
  • 11
15 votes
1 answer
1k views

Krein Milman theorem without the axiom of choice

The Krein-Milman theorem asserts that in a locally convex topological vector space, a nonvoid compact convex subset is the closed convex envelope of its extreme points. But I would like to know when ...
Paul-Benjamin's user avatar
9 votes
1 answer
2k views

Dual or pre-dual of BV

Was there any relevant work to determine the dual (or more likely the predual) of the space of bounded variation functions $BV(\mathbb{R}^n)$ (I recall the definition : a function in $L^1(\mathbb{R}^n)...
Paul-Benjamin's user avatar
2 votes
2 answers
881 views

Uniformly bounded operator family and pointwise convergence

Let $1 \leq p < \infty$ be fixed and let $\Omega \subseteq \mathbb{R}^n$ be open. Let $(Q_n)_{n \in \mathbb{N}}$ be a uniformly bounded family of operators on $L^p(\Omega)$, i.e. there exists $C>...
Matthias Ludewig's user avatar
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 ...
Janak's user avatar
  • 213
2 votes
0 answers
448 views

Lebesgue point and regularity of functions

A known theorem says that for $f \in L_{loc}^1(\mathbb{R}^d)$, almost every point is a Lebesgue point. I know too a theorem saying that for $f \in W_{loc}^{1,p}(\mathbb{R}^d)$ , every point is a ...
user62319's user avatar
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 ...
Manny Reyes's user avatar
  • 5,407
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 ...
Alexander Shamov's user avatar
0 votes
0 answers
148 views

existence of locally translation-invariant Borel measure on Frechet manifolds

It is well known that the only locally finite, translation-invariant Borel measure on an infinite-dimensional, separable Frechet space is the trivial measure. I am wondering about an analogous ...
Erik Curiel's user avatar
2 votes
1 answer
756 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 ...
yada's user avatar
  • 1,773
3 votes
2 answers
470 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 ...
Transcendental's user avatar
5 votes
1 answer
297 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 ...
Transcendental's user avatar
6 votes
0 answers
365 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 ...
Transcendental's user avatar
4 votes
1 answer
819 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 almost-...
Transcendental's user avatar
1 vote
0 answers
205 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 $\...
John's user avatar
  • 11
3 votes
0 answers
119 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 ...
MMML's user avatar
  • 107
18 votes
1 answer
11k 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 ...
polmath's user avatar
  • 321
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 $f(x)=e^{-|x|}$, ...
Anand's user avatar
  • 1,649
6 votes
1 answer
1k 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 ...
Umberto Lardo's user avatar
2 votes
1 answer
104 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 $u<v$...
Braslav's user avatar
  • 67
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 ...
leo monsaingeon's user avatar
4 votes
0 answers
451 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 real-...
Mark Peletier's user avatar
2 votes
2 answers
252 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 $\...
erz's user avatar
  • 5,529