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Measurability of a map involving probability measures

Let $X$ be a metrizable topological space and $\mathscr B_X$ the Borel $\sigma$-algebra on it. Let $\Delta X$ denote the set of probability measures on $(X,\mathscr B_X)$, and let $\mathscr B_{\Delta ...
triple_sec's user avatar
7 votes
2 answers
395 views

Topology on topological spaces

The Gromov-Hausdorff metric makes the set of compact metric spaces into a metric space itself. I am wondering what some natural generalizations there are for arbitrary topological spaces. Namely, is ...
user39598's user avatar
  • 591
-3 votes
0 answers
70 views

Exercise generalizing (related to) Hölder's inequality

I came across this exercise and feel absolutely stuck: Let $p, q, r \in (1, \infty]$ be such that $1/p + 1/q = 1 + 1/r$. Suppose that $F : \mathbb{R}^n \times \mathbb{R}^n \to \mathbb{R}$ satisfies ...
HZA's user avatar
  • 1
2 votes
0 answers
109 views

Uncertainty principle: minimize $\int_{-\infty}^\infty |t| |\widehat{f}(t)|^2 dt$ for $f$ of compact support

This is a question of uncertainty-principle type stemming from Eigenvalue of a convolution and a restriction? Let $f:\mathbb{R}\to \mathbb{R}$ be even, absolutely continuous and supported in $[-\frac{...
H A Helfgott's user avatar
  • 20.2k
0 votes
0 answers
66 views

$L_1$ norm of $f\in L^1(\mathbb{R}^n)$ compactly supported and its change of variable

Let $M\in\mathbb{R}^{n\times n}$ be an invertible matrix, denote its induced linear map on $\mathbb{R}^n$ also by $M$, and let $f\in L^1(\mathbb{R}^n)$ be compactly supported. I am wondering if we can ...
Jens Fischer's user avatar
3 votes
0 answers
52 views

Does there exist a multi-valued "monotone" and "compact" map from a Boolean algebra to the "free" part of $\mathcal{P}(\kappa)$?

This is a follow-up to my previous question, which has a negative answer. Here is the most general version that I'm interested: Does there exist a Boolean algebra $A$, an infinite cardinal $\kappa$, ...
David Gao's user avatar
  • 2,830
4 votes
1 answer
277 views

Eigenvalue of a convolution and a restriction?

Let $\epsilon>0$ be small. Let $\eta(t) = \frac{2\epsilon}{\epsilon^2+(2\pi t)^2}$ (the Fourier transform of $x\mapsto e^{-\epsilon |x|}$). Let $V$ be the space of integrable, bounded functions $f:\...
H A Helfgott's user avatar
  • 20.2k
8 votes
0 answers
243 views
+300

Maps with small fibers between manifolds of equal dimension

The following question is an attempt to revise this one into what I intended. Important revisions are shown in bold. Are there any known examples of a compact Riemannian manifold $M$ with (possibly ...
Matthew Kvalheim's user avatar
3 votes
0 answers
91 views

About BMO space on smooth open bounded domain

Let $\Omega$ be any open domain in $\Bbb R^d$. Define the $\text{BMO}(\Omega)$ space as $$ \text{BMO}(\Omega)= \big\{u\in L^1_{loc}(\Omega)\,\,:\,\, |u|_{\text{BMO}(\Omega)} <\infty \big\}, $$ ...
Guy Fsone's user avatar
  • 1,101
3 votes
0 answers
83 views
+50

Tight upper bound for $m[Q^k - Q^{k+1}]$ for completely positive linear maps

Let $m: \mathcal{L}(\mathbb{R}^{d \times d}) \to \mathbb{R}$ be the function $$ m[H] = \frac{\lambda_{\max}(H[\mathbf{I}])}{\lambda_{\max}(H)}, $$ where $\lambda_{\max}$ denotes the largest eigenvalue....
Ran's user avatar
  • 73
2 votes
0 answers
79 views

Function that is (essentially) a self-convolution but not a multiple of a self-convolution

Call a function $F:\mathbb{R}\to C$ nice if it is of the form $F = f\ast \tilde{f}$, where $\tilde{f}(x) = \overline{f(-x)}$. (Of course nice functions are precisely those whose Fourier transform is ...
H A Helfgott's user avatar
  • 20.2k
2 votes
0 answers
227 views

A deceptively simple regularity problem for functions on the plane

By various meanderings and toying with simpler problems, my current research has lead me to the following quite straightforward question, which I am wholly unable to answer: Consider a twice ...
vmist's user avatar
  • 989
13 votes
1 answer
855 views

Mistake on article about Bohr compactification?

$\DeclareMathOperator\b{b}\newcommand\B{{\operatorname B}}$I wish to get help understanding the content of two theorems of [Iva] that seem mutually contradictory. First some context. Let $\b(\mathbb{R}...
stgo's user avatar
  • 193
5 votes
1 answer
284 views

Codimension zero embeddings and maps with small fibers

Edit: as explained in my comment on alesia's answer, I mistakenly did not ask below the question I intended (due to my misguided efforts to simplify it). Thus, I revised and reposted my question here. ...
Matthew Kvalheim's user avatar
12 votes
1 answer
394 views

Is $X\times X$ homeomorphic to $X$ for a space of probability measures?

Let $\mathcal M_1(S)$ be the (compact, metrizable) space of probability Borel measures on the circle $S=\{z\in\mathbb C: |z|=1\}$ with its weak $*$ topology, so $\mu_n\to\mu$ if and only if $$ \int_S ...
Christian Remling's user avatar
3 votes
0 answers
67 views

Effective action of unbounded operators on subspaces outside their domains of definition

Consider a densely defined, self-adjoint operator $$ H: \mathcal{D} \rightarrow \mathscr{H}. $$ Assume for simplicity that $H$ is nonnegative. We want to effectively restrict this operator $H$ to a ...
Qualearn's user avatar
  • 133
0 votes
1 answer
158 views

Weak convergence of $f(x,e^{itx})$

This is the desired result (what I want to prove): $$f(x,e^{itx})\overset{t\to\infty}{\rightharpoonup}\frac{1}{2\pi i}\oint_{|z|=1}\frac{1}{z}f(x,z)dz \tag{1}$$ Given that $f\in C([a,b]\times\{e^{i\...
Quý Nhân's user avatar
9 votes
1 answer
297 views

What are the points of the algebra of polynomial functions on an arbitrary vector space?

Let $V$ be an arbitrary vector space over some field $\mathbb{K}$ (UPD: of characteristic 0), $V^*=\mathrm{Hom}(V,\mathbb{K})$ its linear dual. Let $\mathrm{Sym}_\mathbb{K}(V^*)$ be the free ...
Dima Roytenberg's user avatar
0 votes
0 answers
46 views

Amenability of locally convex algebras

Let $A$ be an amenable Banach algebra, and let $A_w$ denote $A$ with the weak topology. Clearly, $A_w$ is a Hausdorff locally convex algebra (l.c.a.). Q0: Is $A_w$ amenable as a l.c.a. in the sense ...
Onur Oktay's user avatar
  • 2,605
1 vote
0 answers
85 views

Density of a subset of Schwartz space in the fractional Sobolev space

It is known that the Schwartz space $\mathcal{S}(\mathbb{R}^N)$ is dense in the fractional Sobolev space $H^s(\mathbb{R}^N)$, (where $0<s<1$), as $C_{c}^{\infty}(\mathbb{R}^N) \subset \mathcal{S}...
Nirjan Biswas's user avatar
7 votes
1 answer
294 views

Lower bound on dimension required to disconnect manifold?

This question seems quite classical, but I don't quite know what subarea of topology it falls into. Suppose that removing the set $S$ disconnects the 2-torus $\mathbb{T}^2 = \mathbb{R}^2\diagup\mathbb{...
Ronnie Pavlov's user avatar
3 votes
1 answer
144 views

Jordan plane curve such that $\frac{d(g(x),g(y))}{d(x,y)}\to0$?

Write $g$ as the inverse of $f$. Is there a continuous injective $f:S^1\to C\subset\mathbb{R}^2$ such that $$ \displaystyle\sup_{d(x,y)<r}\dfrac{d(g(x),g(y))}{d(x,y)}\to0 $$ as $r\to0$? If you like,...
Chris Sanders's user avatar
1 vote
1 answer
151 views

Is smoothness preserved under an isometric isomorphism?

Let $(X, \|.\|_1)$ is isometrically isomorphic to $(X, \|.\|_2)$ and $\|.\|_2\leq \|.\|_1$. Assume that $x_0$ is a smooth point of $(X, \|.\|_1)$ and $\|x_0\|_2=1$. According to the definition of a ...
Tuh's user avatar
  • 113
2 votes
0 answers
63 views

Bessel spaces and Triebel Lizorkin

It is known that bessel potential spaces $H^{s,p}$ coincide with Triebel-Lizorkin spaces $F^{s}_{p,2}$ for $s\in \mathbb{R}$ and $1<p<\infty$. Im wondering what can be said por $p=1$ and $p=\...
Guillermo García Sáez's user avatar
1 vote
0 answers
58 views

duality of sobolev spaces. Representation of elements in the dual

I'm trying to understand $(W_0^{1,p} (Ω))^*=W_0^{-1,p^*} (Ω)$, and what a proper representation of its elements is. I understand the basics such as: if $f∈L^{p^*} (Ω)⇒f∈W_0^{-1,p^*}(Ω)$ and the ...
Alucard-o Ming's user avatar
0 votes
0 answers
42 views

questions on stochastic kernels and pushforward operator

Let $f:X \rightarrow \Delta (Y)$ and $g:X \rightarrow \Delta (X)$ be two kernels. For any bounded measurable function $h_Y:Y \rightarrow \mathbb{R},$ define $F(h_Y):X \rightarrow \mathbb{R}$ such that ...
andy's user avatar
  • 1
2 votes
0 answers
157 views

About the 7.3.5. Corollary of the book "Measure Theory" by V.I. Bogachev

According to the 7.3.5. Corollary of the book "Measure Theory" by V.I. Bogachev we have the following result: Let $(X,\tau)$ be a completely regular space and let $\Gamma$ be a family of ...
rfloc's user avatar
  • 637
3 votes
1 answer
343 views

Fundamental group of the grid on $\mathbb{R}^\mathbb{N}$

The grid on $\mathbb{R}^2$ is defined by the set of points such that at most one coordinate is not an integer. With this in mind, e endow $\mathbb{R}^\mathbb{N}$ with the product topology, where $\...
Dominic van der Zypen's user avatar
4 votes
0 answers
90 views

Hölder stability of the PDE $\partial_t u (t, x) = \operatorname{div} \{ a (t, x) \nabla u(t, x) \}$

$ \newcommand{\bR}{\mathbb{R}} \newcommand{\bE}{\mathbb{E}} \newcommand{\bT}{\mathbb{T}} \newcommand{\bP}{\mathbb{P}} \newcommand{\bF}{\mathbb{F}} \newcommand{\cF}{\mathcal{F}} \newcommand{\eps}{\...
Akira's user avatar
  • 835
9 votes
1 answer
295 views

Connected open sets in the topology generated by the collection of connected open sets

Let $(X,\mathcal{T})$ be a connected topological space. Let $\mathcal{T}'$ be the topology on $X$ that is generated by the collection of connected open sets in $(X,\mathcal{T})$. That is, the ...
Calvin Wooyoung Chin's user avatar
0 votes
0 answers
90 views

How to show a point is a weak* -weak continuous for the identity map on $X_1^*$ or on $X_1^{**}$?

I am trying to understand the Remark 3.2 mentioned in the paper titled as "On Weak* -Extreme Points in Banach Spaces" written by S. Dutta and T. S. S. R. K. Rao (http://library.isical.ac.in:...
Tuh's user avatar
  • 113
10 votes
1 answer
659 views

Are there any tests for knowing whether a topological space admits a CW structure?

We know that for n $\ge$ 5, a manifold admits a piecewise linear structure if and only if its Kirby-Siebenmann class vanish and Galewski and Stern showed the existence of a similar invariant to test ...
Tyrannosaurus's user avatar
3 votes
1 answer
132 views

Is the interval topology on ${\cal P}(\omega)/(\text{fin})$ connected?

If $(P,\leq)$ is a poset and $x\in X$, we let $\downarrow x = \{p\in P: p \leq x\}$, and $\uparrow x$ is defined dually. The collection $$\Big\{P\setminus (\downarrow x): x\in P\Big\} \cup \Big\{P\...
Dominic van der Zypen's user avatar
0 votes
1 answer
124 views

Holomorphic functions of certain blow up at origin

Suppose that $D=\{z\in \mathbb C\,:\, |z|\leq 1\}$ and let $f$ be holomorphic on $D\setminus\{0\}$ such that $|f(z)|\leq e^{\frac{1}{|z|}}$ for all $0<|z|\leq 1$ and assume additionally that $\lim\...
Ali's user avatar
  • 4,143
1 vote
0 answers
146 views

integral over the unit sphere of $\Bbb C^n$

Please, is there a way to calculate this integral $$\int_{S_{2n-1}} \frac{e^{a \langle z, \zeta \rangle}}{|z - \zeta|^{\beta}} \, d\sigma(\zeta)$$ where $ z $ is a fixed point in the complex unit ball ...
zoran  Vicovic's user avatar
3 votes
1 answer
90 views

Even covers and collectionwise normal spaces

We call $X$ strongly collectionwise normal if the set $\mathcal{U}_\Delta$ of all neighbourhoods of the diagonal $\Delta_X$ of $X\times X$ is a uniformity. This is equivalent to the property that for ...
Jakobian's user avatar
  • 1,201
2 votes
0 answers
63 views

Lipschitz retraction constant of $B^+$ into $S^+$ in $L^2([0,1])$

In Hilbert space modeled by $L^2([0,1])$ we can define a set $B^+=\{x\in B(0,1): x(t)\geq0 \quad \forall t\in [0,1] \}$ and $S^+=\{x\in S(0,1): x(t)\geq0 \quad \forall t\in [0,1] \}$ where where $B(...
Józef Zápařka's user avatar
0 votes
1 answer
239 views

Are ALL linear functionals on $C[0,1]$ generated by measures? [closed]

Consider derivative of the convolution of a given function $f(\cdot)$ with a fixed $C^\infty$ function $s(\cdot)$, evaluated say at $1/2$. Is there a measure which generates the functional so defined?
H Tomasz Grzybowski's user avatar
4 votes
0 answers
101 views

There is only one reasonable $\sigma$-algebra on the space $\mathcal D'$ of distributions

Consider the space $\mathcal D'(M)$ of distributions on a manifold $M$. Is there a ready reference for the fact that the Borel $\sigma$-algebra (for the strong dual topology) coincides with the weak ...
Pierre PC's user avatar
  • 3,669
1 vote
0 answers
41 views

Why does the Kieboom characterization of shape is restricted only to paracompact spaces?

Borsuk founded shape theory as an extension of homotopy theory, appropriate for spaces with bad local properties. Borsuks definition was applied only to compact metric spaces. Later, this was ...
Emo's user avatar
  • 111
1 vote
0 answers
48 views

What makes the generalized projection different than metric on a Banach space?

I have came across the notion of generalized projection in Banach spaces, introduced by Ya. Alber and has seen many iterative algorithms being solved by using this projection. It helps in finding the ...
PPB's user avatar
  • 85
0 votes
0 answers
53 views

Spectral theory of compact operator for quasi-Banach spaces

Let $X$ be a Banach space and let $Y\subset X$ be a quasi-Banach space (with compact inclusion). Suppose $T:X\to X$ is a compact operator such that $1$ is not its eigenvalue and $T|_{Y}:Y\to Y$ is ...
Liding Yao's user avatar
14 votes
1 answer
500 views

Is there an 'unnatural' topological construction of an algebraically closed field of positive characteristic?

It's well known that while there is a natural topological construction of a nearly algebraically closed field of characteristic $0$, algebraically closed fields of positive characteristic seemingly ...
James E Hanson's user avatar
0 votes
0 answers
62 views

Order-convergence and interval topology on ${\cal P}(\omega)/(\text{fin})$

On any poset $(P, \leq)$ we can consider two different topologies that arise directly from the ordering relation. 1) Order convergence topolog $\tau_o(P)$ : By a set filter $\mathcal{F}$ on $P$ we ...
Dominic van der Zypen's user avatar
5 votes
1 answer
96 views

Preimage of a sublocale by a morphism of locales: description by nucleus?

For completeness of MathOverflow, and to avoid any possible misunderstanding, let me recall the following terminology and facts, which should be standard (experts skip the following 2–3 paragraphs ...
Gro-Tsen's user avatar
  • 32.5k
0 votes
0 answers
77 views

Nice formula for powers of modified Bessel function

Let $K_\nu(z)$ be the modified Bessel function of second kind. I am looking the geometric series $$1+aK_v+(aK_v)^2+(aK_v)^3...$$ I know there are formula for product of two such functions. I would ...
CO2's user avatar
  • 275
0 votes
1 answer
99 views

A question about G-Hewitt spaces

In the paper linked below, S. A. Antonyan gives the following proposition without proof (in fact all results are given without proof). I need a proof of this theorem. If anyone has information on this ...
Mehmet Onat's user avatar
  • 1,367
3 votes
0 answers
158 views

Gowers' dichotomy for quotients

Gowers' dichotomy establishes that every infinite dimensional Banach space contains a closed infinite dimensional subspace that has an unconditional basis or it is hereditarily indecomposable. A ...
M.González's user avatar
  • 4,461
2 votes
0 answers
194 views

Functions such that the *integral* of the Fourier transform is non-negative?

Let $f:\mathbb{R}\to \mathbb{R}$ be in $L^1$, with its Fourier transform $\widehat{f}$ also in $L^1$. What is a necessary and sufficient condition on $f$ so that $$\int_{-\infty}^x \widehat{f}(t) dt \...
H A Helfgott's user avatar
  • 20.2k
2 votes
0 answers
52 views

On distributions and kernels

Let $U\subset\mathbb{R}^{d}$ be an open set and consider $X=\mathbb{R}\times U$. Now, lets consider a smooth (regular) kernel $k_{A}\in C^{\infty}(X\times X)$ and corresponding continuous operator $A:...
G. Blaickner's user avatar
  • 1,429

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