All Questions
Tagged with fa.functional-analysis measure-theory
738 questions
2
votes
1
answer
375
views
Radon-Nikodym derivative in a compact Hausdorff space
Let $X$ be a compact Hausdorff space where $X$ have infinitely many points and the topology is non-discrete, $m$ be a regular probability measure defined on the Borel $\sigma$-algebra of $X$, and $g$ ...
- 307
2
votes
1
answer
282
views
Measurability of a net
Let $(f_\epsilon)_{\epsilon>0}$ be a family of positive measurable functions on $L_p(\mathbb R)$ where $1<p<\infty.$ Assume that the pointwise supremum $f^*(x)=\sup_{\epsilon>0}|f_\epsilon(...
- 1,879
2
votes
1
answer
223
views
Show that $V_G: L^2(G\times G, \mu \times \mu)\to L^2(G\times G, \mu \times \mu)$ defined by $V_G(f)(x,y) = f(xy,y)$ is well-defined
Let $G$ be a locally compact Hausdorff group and let $\mu$ be a right Haar measure on $G$. Then $\mu\times \mu$ (the Radon product of measures) is a right Haar measure on $G \times G$ and we can ...
- 175
2
votes
1
answer
278
views
$\nu$ is a Dirac delta
Let $X$ be an locally compact Hausdorff space and $m$ a positive regular Borel probability measure where $m(Y)$ is 0 or 1 for any Borel set of $X$. Does it necessarily follow that $m$ is a Dirac delta?...
- 103
2
votes
1
answer
354
views
Measurability of integrals with respect to different measures
Let $Y$ be a locally compact Hausdorff topological space (further assumptions like metrizability, separability, etc., may be added if necessary) and let $\mathscr Y$ denote the Borel $\sigma$-algebra ...
- 405
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
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
147
views
Question on density of certain set of matrices
Let $B$ be an invertible real matrix and let $Q=\{A \text{ real}\mid AB^{T} \text{ is symmetric}\}$. Is the subset $S=\{ A \in Q\mid A+A(B^{-1}A)^{2} \text{ is symmetric}\}$ of measure zero in $Q$? I ...
- 286
2
votes
1
answer
77
views
Measurability of random function with values in $C(K,E)$
Let $K \subset \mathbb{R}$ be compact, and let $E$ be a separable Banach space. Further, let $(\Omega, \mathcal{F},\mathbb{P})$ a probability space. I would like to show that a certain a random ...
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
1
answer
197
views
$\Psi$ in finite Wiener–Itô Chaos implies existence of continuous representative on neighborhood of Cameron–Martin space?
Let $(\Theta, H, \mu)$ be an abstract Wiener space, i.e. let $(\Theta, \lVert \cdot \rVert_{\Theta})$ be a separable Banach space, let $(H, \langle \cdot, \cdot \rangle_{H})$ be a separable Hilbert ...
- 150
2
votes
1
answer
183
views
Does set of finitely additive probability measures embed linearly into a strictly convex dual Banach space?
I am trying to better understand a condition that appears in Theorem 1 of this paper.
Let $K$ be a convex and compact subset of a locally convex tvs. The condition is:
$K$ embeds linearly into a ...
- 869
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
115
views
Normalization of Gaussian w.r.t. Gaussian in a Banach space
I would like to compute
$$\int_X \exp\left(-\frac{1}{2}(Au)^2\right)\mathrm d\mu_0(u)$$
with a linear and continuous operator on a Banach space $A:X\to \mathbb R$ (in my case $X=C([0,1])$) and $\mu_0$ ...
- 375
2
votes
1
answer
4k
views
Convergence a.e and $L^1$ boundedness implies convergence in which sense? [closed]
Let $(f_n)$ be a sequence bounded in $L^1 (a,b)$ such that there exists $f$ with $f_n \to f$ a.e.
In which other senses is true that $f_n \to f$? Is is true in $L^1(a,b)$? If there was weak ...
- 201
2
votes
1
answer
891
views
Riesz representation theorem for vector-valued fields
Let $Q$ be a locally compact Hausdorff space, and let $V$ be a topological vector space. Consider the space $X = C_0(Q, V)$ of $V$-valued fields which vanish at infinity. Let $X^*$ denote the dual ...
- 8,512
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
385
views
De la Vallée Poussin criterion on uniform integrability for infinite measures
The de la Vallée Poussin criterion (which is often used in combination with the Dunford-Pettis theorem) is usually formulated for probability measures/finite measures, for example in [Bogachev: ...
- 185
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
139
views
Spaces with atomless independent $\sigma$-sub-algebras
When comparing two sub-$\sigma$-algebras on a probability space $(\Omega,\Sigma,\pi)$, say $\mathcal{X}$ and $\mathcal{Y}$, say that $\mathcal{X}$ is strictly coarser than $\mathcal{Y}$ if the ...
- 123
2
votes
1
answer
274
views
Small ball Gaussian probabilities with moving center
I would like to prove (if possible, otherwise find a counterexample for) the following lemma:
Let $(X,\|\cdot \|_X)$ be a separable Banach space. Additionally, we have a centred Gaussian measure $\mu$ ...
- 375
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
118
views
Control the derivative of a BV function by its symmetric part
Can the derivative of a BV function $f:\mathbb{R}^n\to\mathbb{R}^n$ be controlled by the symmetric part of the derivative $\frac{1}{2}(Df+(Df)^T)$?
- 839
2
votes
2
answers
431
views
References for the Spectral Theorem ( Multiplication Operator Form)
Let $A_1$ and $A_2$ be two commuting self-adjoint (or normal) operators on an infinite-dimensional complex Hilbert space $E$, then there exists a measure space $(X,\mathcal{E},\mu)$,
two functions $\...
- 1,154
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
2
votes
1
answer
2k
views
Countably generated $\sigma$-algebra
Let $(\Omega,\Sigma,\mu)$ be a countably generated probability space. Must $(\Omega,\Sigma,\mu)$ be isomorphic modulo null sets to a standard probability space?
I assume not, so here is a more ...
- 9,720
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 ...
- 1,773
2
votes
1
answer
330
views
functions of bounded variation and gradient vector measure
I want to prove a function of bounded variation on some domain $D\subset R^n$, $f\in BV(D)$, has the property that there is a constant $C$, such that
$$
\lim_{r\rightarrow 0}\frac{C}{r^{n+1}} \int_{...
- 21
2
votes
1
answer
641
views
Fourier transforms of finitely additive bounded measures
Given a finitely additive positive regular bounded measure $\mu$ on ${\mathbb R}^n$ (i.e. a positive linear functional on $C_b({\mathbb R}^n)$), I wonder what can be said about its Fourier transform. ...
- 1,017
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
1
answer
148
views
Is projection of a closed subspace Borel?
Specifically, letting $E$ be a separable infinite-dimensional real Banach space, and $D_2$ in $E\times E$ a closed linear subspace, is then $\{\,x:\exists\,y\,;(x,y)\in D_2\}$ a Borel set in $E\,$? ...
- 3,584
2
votes
1
answer
176
views
Is the measure density condition a necessary condition for bounding the Sobolev norm $W^{n,p}(\Omega)$ by the extremal terms?
Let $\Omega \subseteq \mathbb{R}^M$ be a measurable subset of positive measure. R. A. Adams and J. Fournier in their article have proven that if $\Omega$ satisfies the so-called weak cone property, ...
- 820
2
votes
1
answer
261
views
Qualitative difference between "continuous" and "discontinuous" states on $M(G)$
Let $G$ be a locally compact Abelian group (we can think that $G={\mathbb R}$). Let $C_0(G)$ be the space of continuous functions $u:G\to{\mathbb C}$ vanishing at infinity with the usual $\sup$-norm, ...
- 7,426
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
161
views
Existence of a Borel measurable function
Let $X$ be a compact metric space and $Y\subset X$ be a compact set. Assume that $f_1, f_2: Y \to \mathbb{P}\mathbb{R}^2$ are continuous functions. Let $N \subset \mathbb{P}\mathbb{R}^2$ be a ...
- 133
2
votes
1
answer
164
views
Convergence of measure of products of random unitaries
I'm trying to read Convergence conditions for random quantum circuits by Emerson, Livine, Llyod (https://doi.org/10.1103/PhysRevA.72.060302), arXiv version: (https://arxiv.org/abs/quant-ph/0503210) ...
- 231
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
1k
views
Proving that family of sets has non-empty intersection
Let's say I have an object which can be viewed as family of sets $\mathfrak{S} \subseteq 2^S$, and I want to prove that its intersection is non-empty. What is known already:
$S$ is set of measurable ...
- 105
2
votes
1
answer
310
views
Measure on union of measure spaces and on quotient space
There are two questions about measures bothered me a lot.
Given a set X and a countable covering ${U_i}$ of $X$. Suppose that for each i, there is a measure $m_i$ on $U_i$. Is there a very general ...
- 325
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
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
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
2
votes
1
answer
651
views
Some integrals with respect to a Gaussian measure on a Hilbert space
Assume that $(H,\langle\cdot,\cdot\rangle)$ is a real separable Hilbert space equipped with a Gaussian measure $\mu$ with a mean $m$ and a covariance operator $C$. Let $x\in H$ be a fixed vector. What ...
- 21