All Questions
Tagged with topological-groups measure-theory
20 questions with no upvoted or accepted answers
13
votes
0
answers
421
views
A meager subgroup of the real line, which cannot be covered by countably many closed subsets of measure zero?
Is there a ZFC-example of a subgroup $H$ of the real line $\mathbb R$ such $H$ is meager, has zero Lebesgue measure, but cannot be covered by countably many closed subsets of measure zero in $\mathbb ...
- 41.9k
11
votes
0
answers
263
views
Which results in probabilistic group theory generalize from finite groups to compact Hausdorff groups (and which don't)?
Let $G$ be a finite group. It has been shown that:
If the probability that two randomly selected elements of $G$ generate an abelian group is greater than $5/8$, $G$ is abelian.
If the probability ...
- 495
10
votes
0
answers
272
views
What is the smallest $\sigma$-algebra of reals that is closed under addition of sets?
What is the smallest $\sigma$-algebra $\Sigma\subseteq\mathcal P(\Bbb R)$ containing the open sets and such that if $A,B\in\Sigma$, then $$A+B=\{a+b\mid a\in A,b\in B\}\in\Sigma?$$
I know that neither ...
- 3,011
6
votes
0
answers
110
views
Dependence on Urysohn's Lemma in Cartan's Construction of Haar Measure
This question was posted by someone else on stackexchange three months ago, but no one has answered as of yet:
Cartan's 1940 paper, Sur la mesure de Haar, claims to provide a proof of the existence ...
- 85
6
votes
0
answers
365
views
Approximating a measurable function from a second-countable, locally compact Hausdorff group to a separable Banach space
Let $ G $ be a second-countable, locally compact Hausdorff group and $ B $ a separable Banach space.
We say that a function $ f: G \to B $ is Bochner-measurable if and only if it is the everywhere ...
- 1,396
5
votes
0
answers
202
views
Invariant measure on coset space and integrable functions
Let $G$ be a locally compact abelian group, and $H$ a closed subgroup. Let $C_c(G)$ be the space of continuous, compactly supported complex valued functions on $G$. General theory of Haar measure ...
- 6,180
5
votes
0
answers
214
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On generically Haar-null sets in the real line
First some definitions.
For a Polish space $X$ by $P(X)$ we denote the space of all $\sigma$-additive Borel probability measures on $X$. The space $P(X)$ carries a Polish topology generated by the ...
- 41.9k
3
votes
0
answers
103
views
How wild is the maximal ideal space of the Fourier-Stieltjes algebra of the real line?
The Fourier-Stieltjes algebra of $\mathbb R$ is the set of all sufficiently nice measures on $\mathbb R$. The vector product is convolution of measures. By identifying each measure with its Fourier ...
- 1,955
3
votes
0
answers
740
views
Justification of the convolution operation of $L^1(G)$ functions where $G$ is a LCA group (measurability)
Suppose $G$ is a locally compact abelian Hausdorff group (LCA), and $\lambda$ is the Haar measure on it. We all know the convolution of two $L^1(\lambda)$ functions $f$ and $g$ on $G$ is defined as
$$...
- 960
2
votes
0
answers
187
views
How can the maximal ideal space of the Fourier Stieltjes algebra be non-separable?
I have been asking a fair few (probably elementary) questions about abstract harmonic analysis lately. By means of explanation, I am just feeling around the subject at the moment and trying to build ...
- 1,955
2
votes
0
answers
435
views
Generalized conjugacy classes in (topological) groups
Let $G$ be a topological group. We define an equivalence relation on $G$ as follows:
For $a,b\in G$ we set $a\sim b$ if the following two maps are topologically conjugate:
$$x\mapsto ax,\qquad x\...
- 356
2
votes
0
answers
78
views
Definition of a continuous Gabor frame
I am trying to understand the definition of a Gabor frame and would appreciate some clarification with terminology. Let us begin with the setup: Let $G$ be a locally compact abelian group, and let $\...
- 1,144
2
votes
0
answers
84
views
How to define Lebesgue Integrability of functions assuming values in an arbitrary topological vector space over an arbitrary topological field?
I have already asked this question in this MSE thread, but some people suggested me to ask to the MO community also.
Preliminaries
An algebra of sets in a set $X$ is an $\mathcal{X}\subseteq\mathcal{P}...
- 129
2
votes
0
answers
114
views
Sum-sets of sets of positive measure in the Hilbert cube
Problem. Let $\lambda$ be the standard product measure on the Hilbert cube $[-\frac12,\frac12]^\omega$ and $A,B$ be two $\lambda$-positive Borel subsets of $[-\frac12,\frac12]^\omega$.
Is it true ...
- 41.9k
2
votes
0
answers
102
views
Is this concrete set generically Haar-null?
This question is related to this problem on MO about generically Haar-null sets in locally compact Polish groups but is more concrete.
First we recall the definition of a generically Haar-null set in ...
- 41.9k
1
vote
0
answers
280
views
On measurability of certain group actions on spaces of bounded measurable functions
Let $\mathcal{H}$ be a separable Hilbert space equipped with a strongly continuous unitary representation of a locally compact group $G$. Denote by $\mathcal{L}^{\infty}(H)$ the space of the bounded ...
- 505
1
vote
0
answers
206
views
Extrinsic applications of Haar measure
I am looking for examples of (not necessarily deep) results whose proofs rely on the Haar measure (or rather such that there exists an "elegant" proof involving it). Further, the formulation of such ...
- 111
1
vote
0
answers
115
views
Can Gaussian measure be characterized by unitary representations?
It is well known that Fourier transform switches positive-definite functions with positive measures on a (locally compact topological) group. Further, the positive definite functions can be ...
- 6,259
1
vote
0
answers
220
views
exotic compact group
Let $G$ be compact (and Hausdorff) group, $\mu$ be Haar measure on $G$. Is it always true that $(G,\mu)$ is a standard probability space (Lebesgue-Rokhlin space)? It is so if (a priori not iff) the ...
- 109k
0
votes
0
answers
113
views
The set of measurable functions together with convergence in measure is a completely metrizable abelian topological group
Below we use Bochner measurability and Bochner integral. Let
$(X, \mathcal A, \mu)$ be a complete $\sigma$-finite measure space,
$(E, | \cdot |)$ a Banach space,
$S (X)$ the space of $\mu$-simple ...
- 657