All Questions
155 questions
3
votes
1
answer
100
views
Vague Topologies induced by $C_c$ and $C_0$ are the same on a closed ball of finite Radon measures?
Let $X$ be a locally compact Hausdorff space. Denote $C_c(X)$ and $C_0(X)$ the space of continuous functions with compact support and vanishing at infinity respectively. By Riesz representation ...
- 315
3
votes
1
answer
237
views
Measure theory on abstract Boolean ring
Since a σ-algebra in measure theory is indeed an algebra over $\mathbb{Z}_2$ with addition given by symmetric difference and multiplication given by intersection, does it mean we can put measure on ...
- 43
3
votes
2
answers
949
views
Reference for proof that $C_b^* = rba$
The following theorem seems to have folk status:
The topological dual of the space $C_b(X)$ of bounded continuous functions on a topological space $X$ is isomorphic to the space $rba(X)$ of finite, ...
- 459
3
votes
1
answer
209
views
A particular measure of noncompactness?
I am working on an article based mainly on the notion of Measure of non-compactness, to study a particular type of fixed point theorems.
Let $\mathcal M $ to be the family of all nonempty bounded
...
- 291
3
votes
1
answer
133
views
Restrictions on spectral measure
Given any Borel measure $\mu$ on $\mathbb{R}$, define a map that sends any $f\in C_c(\mathbb{R})$ to $$T_\mu(f)(y)=\int \langle\exp(-i x \lambda),f(x)\rangle\exp(iy\lambda)d\mu(\lambda).$$
Here $\...
- 587
3
votes
1
answer
412
views
Schauder basis of $L^1_{\mathrm{loc}}(\mathbb{R}^n,H)$
$\newcommand{\loc}{\mathrm{loc}}$Let $(\mathbb{R}^n,\mathcal{B}(\mathbb{R}^n),\mu)$ denote the Euclidean space $\mathbb{R}^n$ with its Borel $\sigma$-algebra $\mathcal{B}(\mathbb{R}^n)$ equipped with ...
- 5,405
3
votes
1
answer
274
views
Function square-integrable
Let $f$ be an arbitrary function in $L^2(0,\infty)$ and consider the function
$$(g_f)(y) = \frac{1}{y-x_0} \int_{0}^{\infty} f(x) \left(\frac{xy}{(x^2+y^2+1)}\right)^2 \ dx$$
where $x_0$ is an ...
3
votes
0
answers
222
views
Sets of finite perimeter: intersection with an half space
I have a question regarding sets of finite perimeter. In particular I'm interested to find
$$\mu_{E \cap H_t}, \label{1}\tag{1}$$
where $E$ is a set of finite perimeter in a generic open set $\Omega \...
- 51
3
votes
0
answers
200
views
Largest weak(-like) topology with respect to which continuous functions are dense in the space of Borel functions
Let $X$ denote the space of bounded Borel functions $f\colon [0,1] \to \mathbb{R}$. Let $M$ denote the space of finite Borel measures on $[0,1]$. What is the largest family $F \subset M$ such that for ...
- 1,277
3
votes
0
answers
860
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 ...
- 16.6k
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 ...
- 471
2
votes
3
answers
3k
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 ...
- 1,835
2
votes
2
answers
307
views
Preimage of null sets under a monotone increasing function
Let $I\subseteq \mathbb{R}$ be a closed bounded interval and $f:I \to I$ a monotonic increasing function and $S$ the countable set of points $s$ such that $|f^{-1}(s)| > 1$. Is the following ...
- 113
2
votes
1
answer
117
views
Special density on $L^2$
Let $\Omega\subset\mathbb{R}^N$ be a bounded domain, and $u\in L^2(\Omega)$ with $0\leq u(x)\leq 1$ a.e. on $\Omega$. It is well known that $C^{\infty}_c(\Omega)$ is dense in $L^2(\Omega)$. Because $C^...
- 1,759
2
votes
1
answer
2k
views
Modified Lebesgue differentiation theorem
Let $\Omega\subset \mathbb{R}^n$ an open set and $u:\Omega\to \mathbb{R}$ be a (locally) $L^1$-function. Then it is well known that the Lebesgue differentiation theorem holds: For almost every $x\in \...
- 2,270
2
votes
1
answer
103
views
Sufficient conditions for the space of Radon measure to be a Banach space
Let $\mathcal{X}$ be a Hausdorff space and consider the space of Radon measures with bounded total variation $M(\mathcal{X})$ on $\mathcal{X}$.
Usually, the additional assumptions on $\mathcal{X}$ are ...
- 171
2
votes
1
answer
139
views
Domain of the infinitesimal generator of a composition $C_0$-semigroup
In the paper [1] the following $C_0$-group is presented,
$$
T(t)f(x) = f(e^{-t} x) , \quad x \in (0,\infty) \quad f \in E
$$
where $E$ is an ($L^1,L^\infty$)-interpolation space. In mi case, I'm just ...
2
votes
1
answer
128
views
On the existence of a complicated fractal-like set of finite perimeter
Let $f\in BV(\Bbb R^n)$ be an integer-valued function that maps into $\{0, 1\}$ and is identically $0$ outside some bounded set in $\Bbb R^n$. In particular, $f$ determines a bounded Caccioppoli set $...
- 1,245
2
votes
1
answer
395
views
Existence of integral kernel
I know the following statement ture.
Let $T \in B(L^1(\mathbb{R}^d), L^\infty(\mathbb{R}^d))$, where $B(X, Y)$ denotes all bounded linear operoters from $X$ to $Y$.
Then, $T$ has the integral kernel $...
- 193
2
votes
1
answer
258
views
Control the oscillation of a function by its total variation
Is it possible to control the oscillation of a BV vector field $u:\mathbb R^N \to \mathbb R^N$ at a point $x_0$ by the total variation of $u$?
user140746
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 ...
- 3,245
2
votes
2
answers
230
views
Does the map $f \mapsto \mu_f$ (BV to Lebesgue-Stieltjes measure) behave nicely under function concatenation?
Consider two continuous functions $f,g : [0,1]\rightarrow\mathbb{R}$ of bounded variation, and let $\mu_f, \mu_g : \mathcal{B}([0,1])\rightarrow\mathbb{R}$ be their associated Lebesgue-Stieltjes (...
- 463
2
votes
1
answer
143
views
How to characterize the order convergence in Bochner-integrable functions space?
Let $(\Omega,\Sigma,\mu)$ a finite measure space. We want to characterize the order convergence (for sequences) in Bochner integrable functions space $L^1(\mu,X)$, $X$ Banach lattice.
In $L^p$ we have:...
- 21
2
votes
1
answer
239
views
Injectivity of an integral transform
For a bounded function $F: \mathbb R_{\ge 0} \to \mathbb R$ (not necessarily non-negative), is it true that
$$\int_0^\infty \frac{x^ks}{(s^2+x^2)^{(k+3)/2}} F(x) dx = 0 \text{ for all $s >0$} \iff ...
- 303
2
votes
1
answer
307
views
Box counting dimension of a set and Lipschitz functions
If $f$ is Lipschitz, then the following holds for the Hausdorff dimension:
$$\dim_H f(A) \le \dim_H A.$$
Is the same true for the box counting dimension?
- 839
2
votes
1
answer
328
views
Hausdorff dimension of the graph of a BV function (in 1 dimensional setting)
Let $u: \Omega\subset \mathbb{R} \to \mathbb{R}$ be a function of bounded variation.
Question 1.
How can we prove that the Hausdorff dimension of the essential graph of $u$ equal to $1$?
Question ...
- 839
2
votes
1
answer
263
views
Schwartz space on $\bigcup_{n=1}^CR^n$
I have an application where I need to work with the following idea.
Let the space $\bigcup_{n=1}^C \mathbb{R}^n$ be associated with the metric $d$ such that for $x=(x_1,\cdots,x_n)$ and $y=(y_1,\cdots,...
- 21
2
votes
1
answer
311
views
Differentiation on $[0,1]$
EDIT:
Perhaps a more reasonable question after thinking about the answer I got would have been.
Is there a set $N$ of measure $1-\varepsilon$ and a disjoint partition of that set $N$ with finitely ...
- 536
2
votes
1
answer
140
views
interpret of Picone inequality for non-regular functions
Assume $\Omega \subset \mathbb{R}^N$, $ N>4 $ is open set.
There is a well-known picone identity that says
Let $u,v \in C^2(\Omega)$ satisfy $v>0$ and $-\Delta v \geq 0$ in $\Omega$. The ...
- 371
2
votes
1
answer
577
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 ...
- 717
2
votes
1
answer
474
views
continuous function on the space of probability measures
Let $X$ be a compact space. Let $\mathcal{M}(X)$ be the space of all probability measures on $X$. Denote by $C(X)$ and $C(\mathcal{M}(X))$ the real continuous function on $X$ and $\mathcal{M}(X)$ ...
- 675
2
votes
1
answer
135
views
A non-condensing operator with a power condensing
Let $\alpha$ to be the Kuratowski measure of non-compactness, in a Banach space $E$.
It's very easy to prove that $\alpha (D_1\times D_2)\leq \alpha (D_1)+\alpha (D_2)$, where $D_1$ and $D_2$ are ...
- 291
2
votes
1
answer
196
views
Support of functions in Fourier domain
Let $\mathcal F$ be the Fourier transform. I would like to understand whether being in a Sobolev space implies that the Fourier transform of a function is necessarily supported on a compact ball up to ...
- 23
2
votes
1
answer
240
views
A measure of noncompactness by a convex function
Let $E, \left \| \right \|$ be a Banach space, $\mathfrak{M}_E$ indicate a family of all nonempty bounded subset of $E$, $\mathfrak{N}_E$ the familly of all relatively compact sets, and $Ker \mu=\{X\...
- 291
2
votes
1
answer
1k
views
Pointwise convergence implies uniform convergence?
Let $K$ be an integral kernel of a bounded operator $S:L^2(\mathbb{R}^n) \rightarrow L^2(\mathbb{R}^n) $ defined like
$$(Sf)(x)= \int_{\mathbb{R}^n}K(x,y)f(y)dy.$$
Assume that $K\in C^{\text{bounded}...
- 329
2
votes
0
answers
57
views
Mappings that preserve local or global minimum
In the most general form, I'm interested in any non-trivial results of the following question.
Consider metric space $X$ and $Y$, denote all real valued functions on $X$ and $Y$ as $\mathbb{R}^{X}$ ...
- 275
2
votes
0
answers
29
views
Steiner symmetrization of smooth function on non-simply connected regions
Given a smooth function $u$ defined on $\mathbb{R}^2$, restrict $u$ to a subset $\Omega \subset \mathbb{R}^2$ (possibly not simply connected) foliated by level sets of a smooth function $\psi: \Omega \...
- 434
2
votes
0
answers
175
views
Banach space of vector measures
Let $S$ be a set and $\Sigma$ be a $\sigma$-algebra of subsets of $S$. Let $A$ be a Banach space over the field of complex numbers. A countably additive map $\mu:\Sigma\to A$ is called a vector ...
- 552
2
votes
0
answers
57
views
Is the lattice of bounded Henstock Kurzweil integrable functions countably complete?
The set of HK integrable functions with an integrable upper bound $f$ forms a lattice, and satisfies the MCT and DCT. Does this mean that the lattice is countably complete?
Indexing any countable set, ...
- 1,947
2
votes
0
answers
164
views
(Generalized) Uncentered Maximal Function $\tilde Mf$ in Stein's Harmonic Analysis
It is well known that on $\Bbb R^n$, equipped with the usual Lebesgue measure, the standard Hardy-Littlewood maximal function $Mf(x)$ (with respect to averaging on cubes or balls centered at $x$) is ...
- 1,245
2
votes
0
answers
104
views
Weak convergence rates for integral operators
Suppose $q=\sum_{i=1}^m\pi_i\delta_{x_i}$ is a discrete measure on $\mathbb{R}^n$ and let $q\ast \varphi_\epsilon$ denote the convolution of $q$ with some mollifier $\varphi_\epsilon$, so that $q\ast\...
- 75
2
votes
0
answers
115
views
Showing that for measurable $\Omega \subseteq \mathbb{R}^n$, $L^1(\Omega; C_0(\mathbb{R}^n))$ is separable
Here we're integrating "Banach-valued" functions $u: \Omega \rightarrow C_0(\mathbb{R}^n))$ , and by $u \in L^1(\Omega; C_0(\mathbb{R}^n))$ I mean that
$$\int_{x \in \Omega} \| u(x) \|_{\...
- 21
2
votes
0
answers
89
views
Prove integral inequality for divergence-free vector fields
Let $u$ be a divergence-free vector field $u:\mathbb R^n \to \mathbb R^ n$. Does the following inequality hold?
$$\Big( \int_{\mathbb R^n} |u|^2 dx\Big)^2 \le C\Big(\int_{\mathbb R^n} |u|^2|x|^2 dx \...
- 839
2
votes
0
answers
100
views
What is the weak limit of $f_n \ \mathrm{sign}(f_n - 1)$ if $f_n \to f$ weakly in $L^p([0,1])$?
Let $f_n: [0,1] \to \mathbb R$ be a uniformly bounded sequence in $L^p$. Then there exists a subsequence such that $f_{n_k} \to f$ weakly in $L^p([0,1])$. What is the weak limit of the sequence of ...
- 217
2
votes
0
answers
453
views
Is that correct $\mathbb R^2\cong\mathbb R$ as measurable spaces? [closed]
Is that correct $R^2\cong R$ as measurable spaces?
If we consider $R$ and $R^2$ with Borel $\sigma$-algebras, is there measurable map from $R$ to $R^2$ with measurable inverse?
- 29
2
votes
0
answers
58
views
Absolute continuity of DOS measure for Schrödinger operators
Kotani theory gives roughly that for ergodic operators there is a certain equivalence between absolutely continuous spectrum and an absolutely continuous density of states measure.
I would like to ...
- 21
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 ...
- 21
1
vote
2
answers
209
views
Approximate simple function $f$ by a sequence of continuous functions on $\mathbb{R}^d$ such that $\|f_n\|_\infty\leq \|f\|_\infty$
Let $f=\sum_{i=1}^n c_i 1_{\Delta_i}$ be a simple function on $\mathbb{R}^d$, where $c_i\in\mathbb{C}$. Then we can find sequnces of continuous functions $\{f_k^{(i)}\}$ for each $i=1,\ldots,n$ such ...
- 29
1
vote
2
answers
117
views
If $f\in C([0,\infty))$, does $\delta>0$ and $g\in C^1((0,\delta))\cap C([0,\delta])$ s.t. $g\geq f$ on $[0,\delta]$ and $g(0)=f(0)$ exist?
The question is the following:
Suppose $f : [0,\infty) \rightarrow \mathbb{R}$ is a continuous function. Can I find $\delta \in (0,\infty)$ and a function $g : [0,\delta] \rightarrow \mathbb{R}$ such ...
- 309
1
vote
1
answer
264
views
Is there a version of dominated convergence theorem for local $L^p$ spaces?
Fix $p \in [1, \infty)$. Let $(L^p (\mathbb R^d), \|\cdot\|_{L^p})$ be the Lesbesgue space of $p$-integrable real-valued functions on $\mathbb R^d$. Let $\tilde L^p (\mathbb R^d)$ be the space of ...
- 835