All Questions
Tagged with fa.functional-analysis measure-theory
92 questions
5
votes
2
answers
310
views
Error estimate in the spectral theorem of compact operators on a Hilbert space
Given a compact self-adjoint operator $K$ mapping $L^2(\mathbb{R}^d) \rightarrow L^2(\mathbb{R}^d)$ as $f \rightarrow \int K(x,y) f(y) d\mu(y)$, let us define its eigenvalues $\lambda_i$ and eigen-...
- 2,246
5
votes
1
answer
1k
views
Conditions under which a linear functional on a space of measures must be integration of a function
Let $X$ be a measurable space, and let $M(X)$ be the vector space of finite signed measures on $X$. Are there natural conditions on a linear functional $f:M(X)\rightarrow\mathbb{R}$ that are ...
- 2,130
4
votes
0
answers
119
views
Is the range of a probability-valued random variable with the variation topology (almost) separable?
Let $X$ and $Y$ be uncountable Polish spaces, $\Delta(Y)$ be the space of Borel probability measures on $Y$ endowed with the Borel $\sigma$-algebra induced by the variation distance, and let $g:X\to \...
4
votes
2
answers
374
views
Vague convergence: confusion about the regularity of a signed Radon measure and that of its variation
I'm reading a proof of below theorem from this paper.
Theorem A.3. Let $\Omega$ be a locally compact normal Hausdorff space. Let $\left\{\mu_n\right\} \cup\{\mu\} \subset \mathcal{M}(\Omega)$ and ...
- 657
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 $\...
- 3,738
4
votes
1
answer
412
views
Is a Gaussian measure on a Hilbert space determined by the coarser topology induced by the covariance operator?
I have a basic question about Gaussian measures on a Hilbert space:
Let $\mu$ be a non-degenerate Gaussian measure on a Hilbert space $(H_0,\left\langle \cdot,\cdot \right\rangle_0)$. Then the ...
- 617
4
votes
1
answer
365
views
Lusin Lipschitz approximation in BV and Sobolev space
Theorem 5.34 in Functions of bounded variation by L. Ambrosio, N. Fusco and D. Pallara states that
Let $u \in [BV(\mathbb{R}^N)]^m$. Then there exists a constant $\kappa>0$ such that for every $...
- 839
4
votes
1
answer
228
views
Haar-null union of dense subsets
Let $\{X_i\}_{i \in \mathbb{R}-\{0\}}$ be a set of subsets of a separable infinite-dimensional Fréchet space $X$ and $I$ be uncountable. Moreover, suppose that
(Dense $G_{\delta}$) $X_i$ is a dense ...
- 63
3
votes
2
answers
588
views
On the Fourier inversion formula
For a given function $f\in L^1(\mathbb{R})$, suppose that the
$$\check{f}(x)=\int_\mathbb{R} \hat{f}(\zeta)e^{2\pi i\zeta x}d\zeta$$
almost every where converges in $\mathbb{R}$. Then, can we say that
...
- 4,058
3
votes
1
answer
233
views
A special approximation of the Heaviside function
Is there a $C^m$ approximation $f_\epsilon$ of the Heaviside function such that
$$f_\epsilon(x) = f_1(x/\epsilon) = \begin{cases} 0 & \text{ if } x < 0 \\
1 & \text{ if } x/\epsilon \ge 1
\...
- 131
3
votes
0
answers
144
views
Transitive closure of balanced bounded mass transport
Given two $\sigma$-finite measures $\mu$ and $\nu$ on $\mathbb{R}^n$, write $\mu \sim \nu$ iff there exist countable decompositions $\mu = \mu_1 + \mu_2 + \cdots$ and $\nu = \nu_1 + \nu_2 + \cdots$ ...
- 19.7k
3
votes
1
answer
463
views
"Strongly mutually singular" families of measures, and the set of ergodic measures
Let $(X,\Sigma)$ be a measurable space [which we can assume to be a standard Borel space if we wish].
Let $\mathcal{S}$ be a set of probability measures on $(X,\Sigma)$. [If we wish, we can assume ...
- 708
3
votes
1
answer
274
views
Function square-integrable
Let $f$ be an arbitrary function in $L^2(0,\infty)$ and consider the function
$$(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
1
answer
327
views
$L^1_{\mu}$ as limit
Let $(X,\Sigma,\mu)$ be a $\sigma$-finite measure space. Does there exist a countable set of finite measures $\{\mu_n\}_{n \in \mathbb{N}}$ on $(X,\Sigma)$ such that $L^1_{\mu}(\Sigma)$ can be ...
- 5,405
3
votes
0
answers
233
views
Is it possible to reconstruct the compact space $X$ from the space of measures $M(X)$?
Let $X$ be a compact Hausdorff topological space and $C(X)$ the Banach algebra of continuous functions $u:X\to\mathbb C$ (with the usual $\sup$-norm). It is well-known that the structure of Banach ...
- 7,426
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 ...
- 1,396
3
votes
0
answers
255
views
How can we solve this kind of saddle point problem?
I'm trying to solve a saddle point problem of the following form: Let
$(E,\mathcal E,\lambda)$ be a measure space;
$p$ be a probability density on $(E,\mathcal E,\lambda)$ and $\mu:=p\lambda$
$W$ be ...
- 167
2
votes
1
answer
336
views
Complex Borel measures: does $\mu_n \to \mu$ weakly imply $|\mu|(\Theta) \le \liminf_n |\mu_n|(\Theta)$ for every open subset $\Theta$?
Let
$\Omega$ be a metric space,
$C_b(\Omega)$ the space of all real-valued bounded continuous functions on $\Omega$, and
$\mathcal{M}(\Omega)$ the space of all finite signed Borel measures on $\Omega$...
- 657
2
votes
2
answers
867
views
Decomposition of an abelian von Neumann algebra
Hi, I came across the statement below and I couldn't figure out why it is true. I was hoping someone could explain it or give me a good reference. Thank you in advance.
"Let $\pi$ be a non-degenerate ...
2
votes
0
answers
131
views
Can we conclude $\sup_g\int f_1g\le\sup_g\int f_2g$ from $\int f_1\le\int f_2$ in this situation?
Disclaimer: Please bear with me, the question isn't as complicated as it looks like, but I wasn't able to find any simplification for which no counterexample comes to my find.
Let $(E,\mathcal E,\...
- 167
2
votes
0
answers
199
views
Convergence of the difference quotient of a BV function
Consider a BV function $u \in BV(\mathbb{R}^N; \mathbb{R}^N)$.
What can be said about the difference quotient
$$
\frac{u(x+\epsilon y)-u(x)}{\epsilon}
$$
regarding its convergence as $\epsilon \to 0$...
- 839
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
0
answers
71
views
Is there a function $u \in BV_{loc}(\mathbb{R}^2)$ such that $Du$ is an $s$-dimensional Hausdorff measure restricted to the Koch curve?
Motivated by my previous question Alberti rank-one theorem and irregular jump discontinuities, I'd like to ask the following:
Is there a function $u \in BV_{loc}(\mathbb{R}^2)$ such that $Du$ is an ...
- 839
2
votes
1
answer
168
views
Regarding representation of an outer function
Theorem 2.1 in the book ‘Theory of Hp spaces by Peter. L Duren states that : Any function $f$ analytic on the unit disc belongs to the Nevanlinna class iff it is of the form $\frac{g}{h}$ where $g$ ...
- 111
2
votes
0
answers
240
views
3 questions around the Stone space of the free $\sigma$-algebra on $\omega_1$ free generators
During my studies, I came across several different Stone spaces, e.g.:
(i) The Cantor cube $X=\{0,1\}^{\omega_1}$, which is the Stone space of the free Boolean algebra on $\omega_1$ free generators;
...
- 207
2
votes
1
answer
548
views
Can the characteristic function of a Borel set be approached by a sequence of continuous function through a certain convergence in $L^\infty$?
Want to find $f_n$ a sequence of continuous functions, so that for all Borel regular measure $\mu$, we have
$\int f_n d\mu\rightarrow\int \chi_\Delta d\mu$, as $n$ goes to infinity, where $\Delta$ ...
- 31
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
0
answers
279
views
Relationship between $p$-capacity and Riesz $s$-capacity of a set
What is the relationship between the definitions of $s$-capacity (page 13 here) and $p$-capacity (here) of a set?
Are they equivalent? If not, what inequalities hold? What is the difference (in terms ...
- 839
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
1
vote
0
answers
73
views
Is $L^p_\text{loc} (Y)$ dense in $(L^0(Y), \hat \rho)$?
Below we use Bochner measurability and Bochner integral. Let
$(Y, d)$ be a separable metric space,
$\mathcal B$ Borel $\sigma$-algebra of $Y$,
$\nu$ a $\sigma$-finite Borel measure on $Y$,
$(Y, \...
- 657
1
vote
1
answer
426
views
$L^p$ compactness for a sequence of functions from compactness of cut-off
Fix $p \in [1,\infty)$. Let $f_n:[a,b] \to \mathbb R$, $n \in \mathbb N$, be a sequence of $C^1$ functions. For every fixed $m\in \mathbb N^*$, suppose that the sequence of functions $$\{f_{n}\psi_m(...
- 161
1
vote
0
answers
177
views
Building random homeomorphisms of the torus $\mathbb T^2$
In https://arxiv.org/abs/0912.3423, a family of random homeomorphisms of the circle is constructed. Main Question: Can the construction be generalized to higher space dimensions, e.g. to $\mathbb T^2$?...
- 101
1
vote
1
answer
385
views
$L^p$ compactness for a sequence of functions from compactness of product with cut-off
Fix $p \in [1,\infty)$. Let $f_n:[a,b] \to \mathbb R$, $n \in \mathbb N$, be a sequence of $C^1$ functions. For every fixed $m\in \mathbb N^*$, suppose that the sequence of functions $$\{f_{n}\psi_m(...
- 161
1
vote
0
answers
100
views
Weak estimate for difference quotient of BV function
In an answer to the question Weak Lebesgue spaces and an estimate for BV functions it was remarked that if $u\in BV(\mathbb R^N)$ then there exists a Lebesgue negligible set $F \subset \mathbb R^N$ ...
- 839
1
vote
2
answers
275
views
Again, proving that specific preorder on the set of measurable functions is symmetric
This question is followup to the previous similar question. There I was trying to find good sufficient condition for abstract preorder to be symmetric, but now, as I have found good formalization of ...
- 105
1
vote
0
answers
92
views
Alberti rank-one theorem and reduction of the study of BV function to the two-dimensional case
By Alberti rank-one theorem, could it be possible to reduce the study of a function $u \in BV(\mathbb{R}^N, \mathbb{R}^N)$ to the study of a function $\tilde{u} \in BV(\mathbb{R}^2, \mathbb{R}^2)$? At ...
user123457
1
vote
1
answer
785
views
A question on the Riesz-Markov theorem about dual space of $C_0(X)$
I need to use the version of this theorem about the dual space of $C_0(X)$, which is the set of continuous functions on X which vanish at infinity.
I found in Wikipedia that:
Here the space $X$ is ...
- 43
1
vote
1
answer
178
views
Growth assumption and example of finite (arbitrarily small) time blow up for ODE
Consider the following ODE initial value problem
\begin{align*}
&\frac{d}{dt}\Phi(t,x) = \boldsymbol{F}(t,\Phi(t,x)), & t \in [0,T], \ \ x \in \mathbb{R}^N,\\
&\Phi(0,x) = x, & x \in \...
- 839
0
votes
1
answer
714
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}$ of ...
0
votes
0
answers
118
views
A measure on the group of homeomorphisms of $\mathbb T^2$
Let us consider the group of measure-preserving homeomorphisms of $\mathbb T^2$ (with transformations identified if they agree almost
everywhere) called $G[\mathbb T^2, \mathcal L^2]$. We shall ...
- 101
0
votes
1
answer
216
views
Complex Borel measures: relation between the total variation norm of a measure and those of its real and imaginary parts
Let $X$ be a metric space and $\mathcal B$ its Borel $\sigma$-algebra. For $B \in \mathcal B$ we denote by $\Pi(B)$ the collection of all finite measurable partitions of $B$, i.e.,
$$
\Pi(B)=\left\{\...
- 657
0
votes
0
answers
98
views
Does weak $L^2$ approximation implies $L^2$ approximation under a condition similar to convexity?
(Cross posted from Math StackExchange: Does weak $L^2$ approximation implies $L^2$ approximation under a condition similar to convexity?)
Assume $(\Omega, \mu)$ is a probability space. Consider a ...
- 2,830