All Questions
Tagged with fa.functional-analysis measure-theory
738 questions
9
votes
1
answer
2k
views
Alternative proof of a theorem of Riesz
My question is not research level, but I have not received any feedback on Mathstack; so I am posting it here. I am aware of the traditional proof of the Riesz Theorem that relates linear functionals ...
- 263
4
votes
1
answer
2k
views
Operator topologies on $L^{\infty}(X,\mu )$
Let $(X,\mu )$ be a measure space. Then, $L^2(X):=L^2(X,\mu )$ is a Hilbert space in the usual way and we may view $L^{\infty}(X):=L^{\infty}(X,\mu )$ as a subalgebra of bounded operators on $L^2(X)$ ...
- 1,270
9
votes
2
answers
1k
views
On the definition of "almost-everywhere" for non-complete measure spaces
If $(X,\mathcal{B},\mu)$ is a (non-necessarily complete) measure space, we can give two different notions of a property $P(x)$ that is true almost-everywhere :
(D1) There is a measurable set $A$ ...
- 549
12
votes
1
answer
1k
views
What is the structure associated to almost-everywhere convergence?
Let $M(X)$ be the vector space (actually it's an algebra) of all equivalent classes of measurable functions $X\to \mathbb{C}$ (where $X$ is a measured space) modulo equality almost-everywhere.
One ...
- 549
0
votes
0
answers
76
views
Measure on infinite dimesional $L^p$ space relating size in norm to size in measure
Let $A$ be a bounded set in an infinite dimensional $L^p$ space. Fix an $\epsilon>0$. Is there a Borel measure $M$ such that
$$ M(B(x,\epsilon)) \geq C, \quad \forall x \in A$$
for some $C>0$ ...
- 111
0
votes
0
answers
46
views
The Minkowski $(N-1)$- dimensional upper constant of a closed curve?
Let $\Omega\subset \mathbb R^N$ be open bounded smooth boundary. Let $S\subset \Omega$ be a $N-1$ rectifiable set with $\mathcal H^{N-1}(S)<+\infty$. It is well know that if $S$ is not closed, then ...
- 679
1
vote
1
answer
114
views
Reference request: regularity of functionals on the space of probability measures
Let $\mathcal M=\mathcal M(\mathbb R^d)$ be the space of finite measures on $\mathbb R^d$, and $\mathcal P=\mathcal P(\mathbb R^d)\subset\mathcal M$ be the space of probability measures. Let $F:\...
- 1,835
5
votes
1
answer
611
views
The topological duals of spaces of finite measures
In volume 1 of "Linear Operators", Dunford and Schwartz say that (footnote F1, page 374)
"No completely satisfactory representation for the conjugate space of $ba(S, \Sigma)$, $ca(S, \Sigma)$ or $...
- 5,407
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
1
vote
1
answer
165
views
Decomposition of $L^2$-spaces and singular measures
If $\langle \Omega, \mathfrak{F}, \mathbb{P}\rangle$ is a measure space and $L^2$ is the corresponding $L^2$ space and
$$
K\oplus K^{\perp} \cong L^2(\mathfrak{F},\mathbb{P}).
$$
Then let:
$$
\...
3
votes
0
answers
245
views
Lawvere's 'Categories of space and of quantity" - the universal coefficient theorem
This is a continuation of this question about the paper Categories of Space and of Quantity by W. Lawvere.
As intuitively clear by the very broad (and tentative) definitions suggested by W. Lawvere, ...
- 10.5k
6
votes
0
answers
796
views
What is the Banach dual of the Bochner space $L^\infty(\Omega;X)$?
Suppose $\Omega$ is a $\sigma$-finite measure space (I'm happy to take $\Omega = \mathbb{N}$) and let $X$ be a Banach space. It's pretty well known that the Banach dual of $L^\infty(\Omega)$ can be ...
user14166
2
votes
1
answer
4k
views
Convergence a.e and $L^1$ boundedness implies convergence in which sense? [closed]
Let $(f_n)$ be a sequence bounded in $L^1 (a,b)$ such that there exists $f$ with $f_n \to f$ a.e.
In which other senses is true that $f_n \to f$? Is is true in $L^1(a,b)$? If there was weak ...
- 201
4
votes
1
answer
283
views
Absolutely continuity in variation of constant formula
We are talking here about the initial value problem on some Hilbert space $H$
$$y'(t)=Ay(t)+f(t), \\ y(0)=y_0 \in D(A).$$(Problem 1.13 in the reference)
Then $y(t)=e^{At}y_0 + \int_0^t e^{A(t-s)}f(s) ...
- 43
4
votes
2
answers
801
views
Measure theory and Probability Theory on generalizations of topological spaces
I think this MSE thread is more suitable for the MO community, so I copy it here.
Given a set $X$ and a topology $\tau$ on $X$ the definition of the Borel $\sigma$-algebra $B(X)$ makes use of the ...
- 1,773
3
votes
1
answer
463
views
"Strongly mutually singular" families of measures, and the set of ergodic measures
Let $(X,\Sigma)$ be a measurable space [which we can assume to be a standard Borel space if we wish].
Let $\mathcal{S}$ be a set of probability measures on $(X,\Sigma)$. [If we wish, we can assume ...
- 708
1
vote
1
answer
237
views
Could I affirm that $f$ is not identically 0?
Consider the following situation: Let $\Omega =l^{\infty}(\mathbb{R})$ be the
space of all bounded sequences of real numbers. We will consider in $\Omega$ the metric:
$
d(x,y)=\sum_{i\geq 1}\frac{|...
user11178
2
votes
1
answer
651
views
Some integrals with respect to a Gaussian measure on a Hilbert space
Assume that $(H,\langle\cdot,\cdot\rangle)$ is a real separable Hilbert space equipped with a Gaussian measure $\mu$ with a mean $m$ and a covariance operator $C$. Let $x\in H$ be a fixed vector. What ...
- 21
2
votes
1
answer
278
views
$\nu$ is a Dirac delta
Let $X$ be an locally compact Hausdorff space and $m$ a positive regular Borel probability measure where $m(Y)$ is 0 or 1 for any Borel set of $X$. Does it necessarily follow that $m$ is a Dirac delta?...
- 103
1
vote
1
answer
227
views
Formula for an integration on $\mathbb{Q} \cap [0,1]$
In order to work with functions defined on $\mathbb{Q} \cap [0,1]$ I would like to define an adapted "integration" formula on this set. I though that following definition could be interesting:
$$ \...
- 1,199
4
votes
1
answer
495
views
Weil's Haar measure construction from below
Weil's construction of a Haar measure on a locally compact group rests on approximating a function from above by sums of translates of another function.
I would need to know something similar for an ...
user1688
8
votes
1
answer
2k
views
Definitions of Hilbert Bundles
I have some doubts regarding definitions and conventions on Hilbert Bundles. Some authors like Peter Kuchment (Floquet Theory for Partial Differential Equations) and Serge Lang (Differential and ...
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
2
votes
1
answer
351
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 ...
- 395
4
votes
1
answer
364
views
$\forall g\in L^2(\Omega)$ exists $g_n\in H_0^1(\Omega)$ and $\epsilon>0$ s.t $g_n(x)\to g(x),\,a.e$ and $|g_n(x)|\leq |g(x)|+\epsilon$
I asked this question on MSE here some time ago, but I couldn't get an answer. There was a suggestion in the comments for a counterexample using a fat Cantor set, but I couldn't show a contradiction ...
- 261
4
votes
0
answers
185
views
A strongly open set which is not measurable in the weak operator topology
Let $H$ be a non-separable Hilbert space and $\{e_i\}_{i\in I}$ be an orthonormal basis for $H$. Let $J$ be a uncountable proper subset in $I$.
Let us put $$E=\{x\in B(H): \lVert xe_j\rVert <1: \...
- 4,058
3
votes
1
answer
199
views
A linear functional on $C(K)^*$ continuous on each $L_1(\mu)$
I asked this at math.stackexchange, but nobody answered.
Let $K$ be a (Hausdorff) compact topological space, ${\mathcal C}(K)$ the usual Banach space of continuous functions $x:K\to{\mathbb C}$, ${\...
- 7,426
5
votes
1
answer
601
views
Invariant probability on a unit ball of a Banach space
Let $\Gamma$ be a discrete group acting on an infinite-dimensional Banach space $X$ by linear isometries.
Is there a probability measure (non-atomic, not supported on a finite dimensional subspace) ...
- 53
1
vote
2
answers
530
views
Is there a name for this metric on a Borel sets
Consider a finite measure space $(X,\Sigma,\mu)$.
Consider the function $d:\Sigma \times \Sigma \to [0,1]$ given by
$$d(\sigma_1,\sigma_2) = \mu \left\{ (\sigma_1^c \cap \sigma_2) \cup (\sigma_1 \cap \...
- 614
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
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 ...
- 9,712
8
votes
1
answer
622
views
Completion of spaces of measures w.r.t. weak norms
For the sake of concreteness denote by $M_0(X)$ the linear space of all signed Borel measures $\sigma$ with $\sigma(X)=0$ on some metric space $(X,d)$ and fix some base point $x_0\in X$. On this space ...
- 12.7k
2
votes
1
answer
183
views
Visualizing ANOVA Decomposition [closed]
Let $f \in L^2[0,1]^d$ be a measurable function where $d \in \mathbb{N}$. For a given subset $u \subseteq D := \{1,2,\ldots,d\}$ consider the projections $P_u : L^2[0,1]^d \to L^2[0,1]^{|u|}$ given ...
- 121
2
votes
0
answers
168
views
Interchanging integrals and continuous linear forms in RKHS
I am reading Reproducing kernel Hilbert spaces in probability and statistics by A Berlinet, C Thomas-Agnan.
In Chapter 5 INTEGRATION OF $\mathcal{H}$-VALUED RANDOM VARIABLES they write One of the ...
- 619
1
vote
0
answers
106
views
Can Gradient be controlled by Curl and Divergence in Morrey spaces
In $L^p(\mathbb{R}^3)$, it holds for $1< p< \infty$ and $\mu\in C^\infty_0(\mathbb{R}^3)$,
$$\|\nabla\mu\|_p\leq C \left( \|\operatorname{div} \mu\|_p + \|\nabla\times\mu\|_p \right).$$
So, how ...
- 11
2
votes
0
answers
69
views
Doubts regarding pre-compactness of bounded sequence of measure valued functions
Given:
$\phi(x,\lambda) : \Omega$X$\mathbb R \to \mathbb R^{n}$ be a Caratheodory vector such that for each $M \gt 0 $, $\alpha_{M}(x) = max_{|u| \leq M} |\phi(x,u)| \in L^{2}_{loc} (\Omega)$ .
...
- 121
0
votes
1
answer
193
views
$\int_{R^2}\varphi(x)d\mu(x)=0$ $\Leftrightarrow$ $\sum_{n\in \mathbb Z^2} d\mu(x-2\pi n)=0$
Let $\mu$ be a finite measure supported by $\Gamma $ (a smoth finite curve) and absolutely continuous with respect to the length measure on $\Gamma$ such that $\Gamma \cap (\Gamma+x)$ is a finite ...
3
votes
0
answers
225
views
Sigma algebra generated by SOT versus of sigma algebra generated by WOT
Let $H$ be a non-separable Hilbert space. Let us denote $B_s$ ($B_w$), by the sigma algebra generated by the strong operator topology (weak operator topology) on $B(H)$.
Question: Is $B_s$ the same ...
- 4,058
5
votes
1
answer
289
views
Operator-valued measurable functions
Let $H$ be a non separable Hilbert space and $\Omega$ be a measurable space.
Naturally, we say that $f:\Omega\to B(H)$ is $w$-measurable if $f^{-1}(O)$ is measurable for any open set $O$ in the weak ...
- 4,058
3
votes
0
answers
96
views
Moment Sequence in l²
I have the following problem/question:
For which finite regular complex measures $\mu$ is the moment sequence
$$
\left(\int_{[-1,1]}t^k\,d\mu\right)_{k\in\mathbb N}
$$
a member of $\ell^2(\mathbb N)$?...
5
votes
1
answer
386
views
Lower semi-continuity of the Hellinger-Fisher-Rao distance
I am currently working on unbalanced optimal transport, where the Hellinger (or sometimes Fisher-Rao) distance
$$
H^2(\rho,\mu)=\int_{\Omega}\left|\sqrt{\frac{d\rho}{d\lambda}}-\sqrt{\frac{d\mu}{d\...
- 5,380
4
votes
1
answer
313
views
Invariant subspaces are reducing subspaces in $L^2(\mu)$; where $\mu$ is a singular measure w.r.t Lebesgue measure
I have already posted this question on math.stackexchange but didn't get any answer. I hope this is the right place to ask this question.
Recently I was reading a book "Operator Function and system" ...
- 207
3
votes
1
answer
298
views
Averaging measurable functions over amenable group actions
Let $G$ be an amenable group acting on a space $X$.
Amenability means there is a $G$-invariant mean on $L^\infty(G,{\mathbf R})$.
Given a bounded function $f\colon X\to {\mathbf R}$ one can use the ...
- 10.4k
3
votes
1
answer
259
views
Constructing an additive set function from on a non-additive one
repost from math.se.
I was trying to generalize some results from measure theory to functions that are "almost" measures but not additive. Then, I thought it could be interesting to do this in a ...
- 3,549
8
votes
1
answer
360
views
Can we recover a topological space from the collection of Borel probability measures living on it?
Let $(X, \tau)$ be a topological space, and $\mathcal{P}(X, \tau)$ be the Borel probability measures living on $X$. Can we recover $(X, \tau)$ from $\mathcal{P}(X, \tau)$?
user76098
6
votes
1
answer
243
views
Existence of a measurable map between metric spaces
Let $X$ and $Y$ be separable complete metric spaces (if necessary, they may be assumed to be compact). Let $R\subset X\times Y$ be a closed subset such that the projection of $R$ to $X$ is onto.
Is ...
- 21.8k
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
313
views
Spherical decreasing rearrangement on the sphere
On $\mathbb{R}^n$, we have the concept of spherically decreasing rearragement of a function, which means, given a function $f$, one can design a radial and decreasing function $f^*$ such that $\Vert f^...
- 21
4
votes
1
answer
78
views
Bound for the generalised Rényi dimension of a measure
If $\mu$ is a measure on $\mathbb{R}^d$, and for each $r>0$ we let $\mathcal{M}_r$ denote the set of all ``cubes'' of the form $[j_1r,(j_1+1)r) \times \cdots \times [j_dr,(j_d+1)r)$ for $j_1,\ldots,...
- 6,206
3
votes
2
answers
579
views
Example of measure of non-compactness
I can't understand the following example of measure of non-compactness, which was given in this article.
Definition: A non-negative function $\phi$ defined on the bounded subsets of $X$ will be ...
- 153