All Questions
Tagged with fa.functional-analysis measure-theory
738 questions
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
77
views
Convergence of some object depending on functions with compact support
Let $G$ be a locally compact group with unimodular Haar measure $\mu$. We consider the Hilbert space $\mathscr{H}:= L_{\mu}^2(G)$ together with the unitary representation $\pi : G \to U(\mathscr{H})$ ...
- 429
2
votes
1
answer
203
views
Non-uniqueness in Krylov-Bogoliubov theorem
So apparently the Krylov-Bogoliubov theorem says that every continuous function $f:X\to X$ on a compact metrizable space $X$ has an invariant probability measure $\mu$.
Of course, if $X$ is just a ...
- 24.8k
2
votes
1
answer
140
views
Measurability of $T \to \Pi_{ker T}$ w.r.t. SOT
I came across the following technical question, to which I could not - after some time of thinking - find an answer:
Let $\mathcal{U},\mathcal{H}$ be two real (in general infinite dimensional) ...
- 133
1
vote
1
answer
284
views
Recover norm from integral
I am given the following expression where $f \in L^2(\mathbb{R}^2, \mathbb{R}^{2 \times 2})$
$$\int_{\mathbb{R}} \int_{\mathbb{R}} \langle g(x), f(x,y) h(y)\rangle dx dy.$$
The functions $g$ and $h$ ...
user121558
0
votes
2
answers
494
views
Semifinite measure and spectral theorem
Let $H$ be a complex Hilbert space (not necessary separable).
Spectral Theorem: Let $A_1$ and $A_2$ be two commuting normal operators, then there exists a measure space $(X,\mathcal{E},\mu)$,
two ...
- 1,154
4
votes
2
answers
175
views
Geometric mean of positive measures
Let me given with an obvious example. Let $\Omega\subset{\mathbb R}^n$ be an open domain. If $f,g\in L^1(\Omega)$ and $f,g\ge0$, then $\sqrt{fg}\,\in L^1(\Omega)$.
Now let me replace the absolutely ...
- 52.3k
2
votes
0
answers
212
views
Bilinear form representation
It is known, that the following result holds: let $X,Y$ be compact sets in $\mathbb{R}^N$ and $\mathbb{R}^M$ respectively and let $B:C(X)\times C(Y)\to\mathbb{R}$ be a continuous bilinear form ($C(X)$ ...
user114232
2
votes
1
answer
336
views
Regarding characterisation of outer functions in a Hardy space
Please see the definition of Hardy spaces on the unit disc here. This is regarding outer functions on a Hardy space. I know that outer functions can have no zeroes in the open unit disc since it is ...
- 111
1
vote
0
answers
51
views
Extension of $\sigma$-additive vector measures on a ring and the relationship between the corresponding total variation functions
Let
$\Omega$ be a set
$\mathcal S\subseteq2^\Omega$ be a set with $\emptyset\in\mathcal S$
$\mathcal S_{\text{loc}}:=\left\{A\subseteq\Omega:A\cap S\in\mathcal S\text{ for all }S\in\mathcal S\right\}$...
- 167
1
vote
1
answer
87
views
Function is almost everywhere 1 w.r.t. sequence of regular Borel probability measures
Let $\epsilon>0$ be given. Let $Y$ be a compact, Hausdorff space and let $U\subseteq Y$ be an open subset. Assume that $(\mu_n)_{n\in\mathbb{N}}$ is a sequence of regular Borel probability measures ...
- 11
2
votes
0
answers
164
views
An operator valued Egoroff's theorem
The following statements suggests $B(H)$-valued Egoroff's theorem when $H$ is a separable Hilbert space. Probably it will be hold even if a von Neumann algebra $M$ whose predual is separable is ...
- 4,058
2
votes
0
answers
116
views
Density of $C^0(\Bbb R^{n}\times (0,T))$ and $C^{\infty}_c(\Bbb R^{n}\times (0,T))$ in $L_{p,q}(\Bbb R^n_T)$
Let the space $L_{p,q}(\Bbb R^n_T)$ be defined as the set of all measurable $f:\Bbb R^{n}\times(0,T)\to\Bbb R$ such that $||f||_{p,q}<\infty$, where
$$
||f||_{p,q}:=\left(\int_0^T\left( \int_{\Bbb ...
- 1,245
2
votes
1
answer
298
views
Regarding outer functions again
Consider the Hardy space $H^p, 0<p\leq\infty$ (defined here).
It is said that given any two outer functions $x_1$ and $x_2$ in $H^p$, there exists $a_1$ and $a_2$ in $H^\infty$ such that $a_1x_1=...
- 111
2
votes
1
answer
168
views
Regarding representation of an outer function
Theorem 2.1 in the book ‘Theory of Hp spaces by Peter. L Duren states that : Any function $f$ analytic on the unit disc belongs to the Nevanlinna class iff it is of the form $\frac{g}{h}$ where $g$ ...
- 111
6
votes
0
answers
4k
views
Interchange of supremum and integral
Let $f : X \to Y$, $X \subset R^n$, $Y$ Banach space, $g : X \times Y \to R \cup \{ \infty \}$, $L^n$ the n-dimensional Lebesgue measure.
Are there some results under which the following interchange ...
- 303
4
votes
1
answer
253
views
Regarding outer functions
Please see the definition of Hardy spaces on the unit disc here. Let $0<p\leq\infty$. Let $f\in H^p$ with $\|f-1_e\|_p<1$ (Where $1_e$ Is the constant function one). Then is $f$ an outer ...
- 111
2
votes
0
answers
58
views
Absolute continuity of DOS measure for Schrödinger operators
Kotani theory gives roughly that for ergodic operators there is a certain equivalence between absolutely continuous spectrum and an absolutely continuous density of states measure.
I would like to ...
- 21
1
vote
1
answer
178
views
Bochner integrability within a subspace
Let $(H,||\cdot||_H)$ be a Banach space and $K$ a (not necessarily closed) subspace. Suppose that $K$ is a Banach space under another norm $||\cdot||_K$, which satisfies
$$||x||_H\leq ||x||_K$$
for ...
- 1,903
1
vote
1
answer
201
views
Existence of a certain norm on space of measurable functions
Suppose $X$ is a measure space with measure $\mu$. Given a strictly increasing continuous (or sufficiently nice) function $\phi:[0, \infty)\to [0, \infty)$ with $\phi(0)=0$. Is it true that we can ...
- 13
6
votes
1
answer
2k
views
About weak convergence of probability measure
Suppose $\mu_j$ is a sequence of measures on $\mathbb{R}$. By the definition of weak convergence of measures, $\mu_j$ weak converges to $\mu$ means that for any bounded continuous function $f$, there ...
- 71
16
votes
4
answers
7k
views
Good book for measure theory and functional analysis
I have taken advanced courses both in measure theory and also in functional analysis (Banach and Hilbert spaces, spectral theory of bounded and unbounded operators, etc.)
The connections between the ...
0
votes
1
answer
268
views
Spectral Theorem, $AB = BA \implies B\Phi(f) = \Phi(f)B$
I'm studying the spectral theorem as appears in Reed and Simon's Functional Analysis.
Assume we have constructed the continuous functional calculus for a self adjoint bounded operator $A$ on a ...
- 181
1
vote
1
answer
194
views
Proof that the subspace of signed measures integrating d(x,e) is closed
Let $\mathcal{M}(S)$ be a space of finite signed measures on a metric space $S$ ($=\mathbb{R}^2$ in my case) equipped with the total variation norm. Let
$\mathcal{M}_1(S)=\{\mu \in \mathcal{M}(S):\...
- 13
4
votes
1
answer
295
views
Literature request: Functional capacities
Are the results of the following book (in French) covered in English in a book or in an article, and if so, could you please provide a reference?
C. Dellacherie, Ensembles analytiques, Capacités, ...
- 53
2
votes
1
answer
155
views
Two questions related to Dirichlet spaces and Sobolev spaces
I want to ask a question that arises from reading this paper.
Let $X$ be a locally compact space which is countable at infinity and let $\xi$ be a Radon measure on $X$. Suppose $V$ is a Hilbert ...
- 65
7
votes
0
answers
549
views
Counter-example to the completeness of the Wasserstein metric
$\newcommand{\P}{\mathcal{P}}$
Let $(E,d)$ be a complete metric space, let $\P(E)$ be the set of all probability measures on $(E,\mathcal{B}(E))$. Let $W_d$ be the $1$-Wasserstein (Kantorovich) ...
- 931
1
vote
1
answer
139
views
Compactly supported functions and projections
Let $\Omega$ be an open subset of $\mathbb{R}^n$ and take a family of continuous compactly supported functions $f_n$ on $\Omega$ normalized to one (in the $L^2$ sense).
Then, these functions span a ...
- 501
7
votes
0
answers
478
views
Characterizing the sum $L^1 + L^\infty + L^{1,\infty} + L^{\infty, 1}$ of iterated Lebesgue spaces "by duality"
For the usual Lebesgue spaces $L^p (\mu)$ ($p \in [1,\infty]$) on a ($\sigma$-finite) measure space $(X,\mu)$, it is well-known that one has the characterization
$$
L^p (\mu) = \left\{f : X \to \Bbb{...
- 734
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
9
votes
1
answer
636
views
Is there a characterization of the Hausdorff measures?
It is known that there is a unique measure on the Borel $\sigma$-algebra of $\mathbb{R}^n$ such that the measure of the rectangle $\prod_i [a_i,b_i[$ is $\prod_i (b_i-a_i)$. This is the Lebesgue ...
- 1,035
6
votes
1
answer
1k
views
Tensor product of measure spaces
For a compact topological space $X$, denote by $\mathcal{M}(X)$ the Banach space of finite signed Borel (Radon) measures on $X$ with the total variation norm. This is canonically isometric to the dual ...
- 9,853
9
votes
1
answer
570
views
Elements of $L^p$ and nice representatives of equivalence classes
Considering $L^p$ $( 1 \leq p < \infty)$ as a normed vector space, each element of $L^p$ is actually an Equivalent class. Take $[f] \in L^p $ as an Equivalent class, What is the Nicest possible ...
- 369
6
votes
1
answer
1k
views
Are Bochner measurablity and Borel measurability compatible?
Let $(X,\mathfrak{M})$ be a measurable space and $E$ be a Banach space and $f:(X,\mathfrak{M})\rightarrow E$ be a function.
Question
Are Borel-measurable condition on $f$ and Bochner-measurable ...
- 337
6
votes
0
answers
388
views
Closedness of a set of measures, where conditional marginals are in closed $\varepsilon$-ball w.r.t. Wasserstein distance
Let $(E,d)$ be a bounded polish space (separable, complete metric space satisfying $\sup_{x,y\in E} d(x,y) < \infty$). By $\mathcal{P}(E)$ we denote the space of Borel probability measures on $E$ ...
- 1,095
1
vote
1
answer
183
views
Diffuse measure space as a product of $[0;1]$ and another diffuse measure space
The title speaks of itself. How far is an arbitrary finite diffuse measure space from being almost isomorphic to a product of $[0;1]$ with another diffuse measure space? What would be reasonable ...
- 1,959
18
votes
1
answer
3k
views
How bad can the second derivative of a convex function be?
One can easily construct an example of a measurable function $f:(a,b)\to \mathbb{R}$ which satisfies the following property:
$$\label{p}\tag{P}
f\notin L^1(I),\ \mbox{for each interval}\ I\subset (a,...
- 409
4
votes
0
answers
585
views
Dual of the space of all bounded functions, $B(X, \mathbb{R}).$
Let $X$ be a non compact separable metric space. Denote
by $B(X, \mathbb{R})$ the set of all bounded real functions
endowed with the sup norm, this is a Banach space. Denote by
$C_b(X,\mathbb{R})\...
- 757
7
votes
0
answers
3k
views
What is vague convergence and what does it accomplish?
For convenience, let's say that I have a locally compact Hausdorff space $X$ and am concerned with probability measures on its Borel $\sigma$-algebra $\mathcal{B}(X)$. Natural vector spaces to ...
- 1,104
1
vote
1
answer
91
views
Design measure, which cannot be factorized as a product of measures
Let $\mathcal{S}_x$ and $\mathcal{S}_y$ be a finite discrete sets, such that
$$
0 < |\mathcal{S}_x| < \infty, \qquad 0 < |\mathcal{S}_y| < \infty, \qquad \mathcal{S}_x \cap \mathcal{S}_y =...
- 181
6
votes
1
answer
2k
views
Convergence of Radon Nikodym derivatives
I apologise in advance if my question is too basic.
Some notation:
$(X,\cal{X})$ denotes a measurable metric space
where $X$ is a metric space and
$\cal{X}$ is the associated Borel sigma algebra.
...
- 757
1
vote
1
answer
287
views
Interpolation between $L^1$ and $L^2$ spaces
I was wondering whether the following interpolation between $L^1$ and $L^2$ spaces is true:
Let $f \in \mathbb{R}^n$ be such that
$$ \alpha_1:= \int_{\mathbb{R}} \left\lVert f(x_1,\cdot,....\cdot) \...
- 175
11
votes
3
answers
3k
views
Dual space of $L^2(\mathbb{R},L^1(0,1))$?
I was wondering what the dual space of $L^2(\mathbb{R},L^1(0,1))$ is? (equipped with Lebesgue measures)
Formally, one would suspect that it is just $L^2(\mathbb{R},L^{\infty}(0,1))$. But this may be a ...
- 175
4
votes
0
answers
123
views
Converse on the rectifiability of products of rectifiable sets
Let $1\leq k\leq m$ and $1\leq l\leq n$ fixed integers, $\mathscr{H}^k$ the $k$ dimensional Hausdorff measure and $E\subset \mathbb{R}^m$. We say that :
(1) $E$ is $k$ rectifiable if there exists $C\...
- 1,616
3
votes
0
answers
473
views
textbook of measure theory abstracted as functional analysis [closed]
Background
I have studied intro functional analysis, probability theory based on measures, and some elementary connection between them e.g. that weak conversion of random variables correspond to weak*...
- 163
3
votes
0
answers
225
views
Defining a trace-class operator with a Bochner integral
I had asked this question previously on Math.StacheExchange but did not get an answer there in several months. This isn't strictly speaking research level mathematics but I hope it is sufficiently ...
- 131
5
votes
1
answer
1k
views
Conditions under which a linear functional on a space of measures must be integration of a function
Let $X$ be a measurable space, and let $M(X)$ be the vector space of finite signed measures on $X$. Are there natural conditions on a linear functional $f:M(X)\rightarrow\mathbb{R}$ that are ...
- 2,130
1
vote
0
answers
96
views
Infimum of equivalent measures
Suppose I have a functional of the form
$$
F(\mathbb{P})\triangleq \int_{\mathbb{R}^d} \int_{\Omega}f(x,\omega)\mathbb{P}(d\omega)m(dx),
$$
where $m$ is the Lebesgue measure and $\mathbb{P}$ is a ...
- 173
6
votes
1
answer
469
views
Poincare Recurrence by Mean Ergodic Theorem
I have a question regarding a confusion from reading the Princeton Companion to Mathematics on the topic of Ergodics Theorems. It is about proving a stronger version of Poincare Recurrence Theorem ...
- 1,245
9
votes
2
answers
548
views
What mode of convergence is this?
I'm interested in a new (to me) mode of convergence which is stronger than convergence in measure/probability. I want to know if it has a name and if it is used much in the literature. I will write ...
- 6,287