All Questions
Tagged with gr.group-theory measure-theory
52 questions
-3
votes
2
answers
195
views
Which self homeomorphisms preserve measure on a torus, apart from affine? [closed]
Which self homeomorphisms preserve measure on a torus, apart from affines? Affine is the composition of rotation and automorphism. Measure is the Lebesgue measure.
- 83
0
votes
1
answer
142
views
Why are the homeomorphisms from the unit circle to the unit circle preserving measure affine? [closed]
Why are the homeomorphisms from the unit circle to the unit circle preserving measure affine? The affine is composition of rotation and continue automorphism.
- 83
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
3
votes
1
answer
119
views
Existence of a bounded measurable subset of $\text{SL}(d,\mathbb R)$ that is Borel isomorphic to $\text{SL}(d,\mathbb R)/\text{SL}(d,\mathbb Z)$?
$\text{SL}(d,\mathbb R)/\text{SL}(d,\mathbb Z)$ has two interesting properties: on one hand it is non-compact, but on the other hand it admits a unique $\text{SL}(d,\mathbb R)$-invariant finite ...
- 321
-1
votes
1
answer
168
views
Følner sequences of the integers
Definition: Let $G$ be a group. For $g\in G$ and a subset $F\subseteq G$ fix
the notation $gF:=\{gf\mid f\in G\}$. A sequence $(F_{i})_{i\in\mathbb{N}}\subseteq G$
is called a Følner sequence if
\...
- 741
3
votes
0
answers
148
views
Can the Banach-Tarski paradox or Tarski's circle-squaring problem be done with hinges?
It is known for both the Banach-Tarski paradox and Tarski's circle-squaring problem that you can finitely partition the starting configuration, then continuously move these pieces (without ...
- 700
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
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
3
votes
0
answers
115
views
Malliavin-Shavgulidze (type) measures on the group of measure-preserving invertible maps on $\mathbb T$?
The Malliavin-Shavgulidze measures on $\operatorname{Diff}^{1}(I)$ (with $I$ an interval of $\mathbb R$) are defined as the image $W_{\sigma} \circ f^{-1}$ of the Wiener measure $W_{\sigma}$ with ...
- 101
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
3
votes
1
answer
361
views
Equivalent definitions of strongly proximal action
Consider the following fragment from the paper "C*-simplicity and the unique trace property for discrete groups" by Breuillard, Kalantar,
Kennedy and Ozawa:
I have two questions:
(1) What ...
- 175
9
votes
1
answer
346
views
Is there a uniform solution of the Ruziewicz problem?
For any integer $n\geq 2$ there is one and only one (up to rescaling) rotation-invariant, finitely-additive measure on the Lebesgue $\sigma$-algebra of $S^n$.
The proof of this statement I'm aware of ...
- 91
9
votes
3
answers
505
views
Non-measurable sets on groups from translation invariance
The most well-known construction of a non-measurable set is the Vitali set. The idea behind Vitali sets is to split up the space (such as $[0,1]$) into equal-sized copies (guaranteed by translation ...
- 163
3
votes
1
answer
237
views
Invariant measure of a subgroup
Let $G$ be an abelian group with a $G$-invariant metric $d$. Let $H$ be a countable dense subgroup of $G$. Let $\mu$ be a non-atomic $\sigma$-finite Borel measure on $G$ that is $H$-invariant. Must it ...
- 59
2
votes
1
answer
156
views
Ergodic decomposition - how does restricting measure effect it? (Choquet Theory)
Suppose that $G$ is a discrete countable group and $\mu$ is an IRS (invariant random subgroup) of $G$: $\mu$ is conjugation invariant as a probability measure on the subgroups of $G$.
Since all the $...
- 181
6
votes
1
answer
291
views
Comparing $X+Y$ and $X-Y$ for independent random variables with values in an abelian locally compact group
Let $G$ be an abelian locally (separable?) compact group with Haar measure $\mu$. Inspired by the interesting proof of A sum of two binomial random variables :
Let $X$ and $Y$ be $G$-valued ...
- 2,101
0
votes
1
answer
64
views
measures on groups without assuming a locally compact group topology
I'm interested in knowing whether there exists any kind of theory for measures on groups without assuming that it's the Haar measure for a locally compact group topology.
- 2,125
2
votes
1
answer
198
views
Equivalence of harmonic measures on hyperbolic groups
Consider a Gromov-hyperbolic group $\Gamma$ and let $\mu$ be a finitely supported probability measure on $\Gamma$. Assume that the support of $\mu$ generates $\Gamma$ as a semi-group, in other words, ...
- 2,090
1
vote
0
answers
106
views
Change variable in integration with symmetry
Not sure if I can ask such fundamental problem here.
Let
$G$ be a finite group, $\sigma \in G$. Consider linear group actions fo $G$ on $\mathbb{R}^n$.
$\sigma(K)=\{x\in \mathbb{R}^n: \sigma^{-1}(...
- 345
5
votes
0
answers
154
views
Continuity of the Green function with respect to the measure
Let $G$ be a finitely generated group and let $\mu$ be a finite measure on $G$. Define the Green function as
$$G(\mu)=\sum_{n\geq 0}\mu^{*n}(e),$$
where $\mu^{*n}$ is the $n$th convolution power of $\...
- 2,090
6
votes
1
answer
424
views
Measure on cosets in a group?
If $\mu$ is the normalized counting measure on a finite group $G$, then $\mu(G)=1$ and $\mu(C)=1/n$ for every coset $C$ of a subgroup of index $n$. Let's ask for the same for infinite groups:
...
- 1,074
4
votes
1
answer
414
views
Finitely Additive Homogeneous Translation Invariant Measure on $\mathcal{P}(\mathbb{R})$
It is known that there exists a finitely additive translation invariant measure on $\mathbb{R}$ that extends the Lebesgue measure. I.e. a function $m:\mathcal{P}(\mathbb{R}) \rightarrow [0,\infty]$ ...
- 297
3
votes
2
answers
541
views
Important examples or applications of lattices in locally compact groups
If $G$ is a locally compact group and $\Gamma $ is a discrete subgroup such that the quotient $G / \Gamma$ carries a finite left $G$-invariant Haar measure, then we say that $\Gamma$ is a lattice in $...
1
vote
1
answer
149
views
Conull subspace containing orbit of an (ergodically acting) group
I should probably start with a warning that this is my first post in this board and that I am sorry, if it is not up to standards. It would be great, if you could let me know how to improve the post.
...
- 13
11
votes
2
answers
578
views
Homeomorphisms vs Borel automorphisms
Let $\mathrm{Homeo}(M)$ and $\mathrm{Borel}(M)$ be the groups of homeomorphic and Borel automorphisms of a space $M$, respectively.
Question: Are $\mathrm{Homeo}(M)$ and $\mathrm{Borel}(M)$ ...
- 1,959
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
12
votes
2
answers
362
views
Are finitely generated amenable groups positively finitely generated?
Let $G$ be a finitely generated amenable group.
Is there a positive integer $n$ such that $n$ random elements of $G$ generate it with positive probability?
Being more formal, note that $G^n$ is ...
- 11.3k
7
votes
0
answers
305
views
Generator of a $\bigoplus_{n=0}^\infty \mathbb{Z}/2\mathbb{Z}$-action
Let $T$ be a measure-preserving action of a group $G$ on a Lebesgue space $X$. That means that $T$ associates an automorphism (i.e. an invertible measure-preserving transformation) $T^g$ of $X$ to ...
- 2,560
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
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 ...
- 41.8k
5
votes
1
answer
630
views
Characterizing residually amenable groups
Let $G$ be a finitely generated group. The amenability of $G$ is equivalent to the existence of a certain "weak measure" on $G$. Is there such a characterization for residually amenable groups as well?...
- 11.3k
7
votes
1
answer
349
views
Can Haar measure fail to be bi-invariant without conjugation shrinking a set?
(This is a slightly reformatted and clarified version of my question from math.SE, since I believe
the answer there is wrong and its poster has not responded to my comment in over two weeks.)
Let $\:...
user5810
9
votes
3
answers
654
views
measure with given push-forwards
Let $X,Y$ be locally compact spaces (in my specific case, they are locally compact groups). Suppose that we are given a measure $\mu$ on $X$ and a finite number of quotient maps $p_1,\ldots,p_n:Y\...
3
votes
1
answer
263
views
Extending Tarski's Theorem on invariant measures
Tarski's Theorem says that if $G$ acts on $X$ and $E$ is a non-$G$-paradoxical subset of $X$, then there is a finitely additive $G$-invariant measure $\mu:2^X\to[0,\infty]$ with $\mu(E)=1$.
I am ...
- 2,555
6
votes
1
answer
716
views
Are there uniformly discrete paradoxical subsets in $\mathbb{R}^3$?
I think there aren't any discrete paradoxical subsets in $\mathbb{R}^2$ (any isometry that mapped a discrete subset into itself would have to either be a glide-reflection, a translation or a rotation ...
- 2,555
23
votes
2
answers
7k
views
What is a Gaussian measure?
Let $X$ be a topological affine space. A Gaussian measure on $X$ is characterized by the property that its finite-dimensional projections are multivariate Gaussian distributions.
Is there a direct ...
- 8,512
2
votes
1
answer
368
views
Borel Group on R [closed]
Last week in class we used the fact that if we have a group within R which is also a Borel Set, then it is either R or meagre. Why is it so? Can you direct me to a proof?
- 397
1
vote
1
answer
874
views
Are all compact groups amenable ?
Wikipedia states that the Haar measure on a compact group is a mean (and that every compact group is amenable). But, obviously, the Haar mesure on the group of unit quaternions cannot be defined on ...
- 3,630
5
votes
1
answer
437
views
Stationary, ergodic measures from the structuralist point of view
Stationary, ergodic measures are a class of objects very familiar to probabilists. In a sense, these are the weakest generalization of the classic case of independent, identically distributed random ...
- 8,512
9
votes
4
answers
1k
views
Symmetries of probability distributions
When talking about a single random variable, knowing only its distribution, the construction of a probability space is quite easy. Namely, let $(X,\mathscr A)$ be a measurable space and let $\mathsf Q$...
- 1,655
6
votes
0
answers
301
views
Generating stationary, ergodic random fields on a homogeneous space
Consider a homogeneous space $M$, which for the sake of concreteness, let's take to be $M = \mathbb R^d$. Fix some space $A$, and consider the space of functions $X = C(M,A)$, along with its Borel $\...
- 8,512
4
votes
0
answers
223
views
A question about measures on groups
Let $G$ be a finitely generated (in my case also amenable) group and $f:G\to[0,1]$. Suppose that there is a finitely additive probability measure $\mu$ on $G\times G$ and a real number $L$ such that $\...
- 6,034
12
votes
3
answers
891
views
Looking for at least one beautiful and not too technical result in asymptotic group theory
We have a student seminar devoted to the problems of asymptotic group theory with some connections to ergodic theory and measure theory in general. Each talk concerns one of the problems of this ...
3
votes
1
answer
401
views
Fubini's theorem and unique mean value
Following the terminology of Rosenblatt, I will say that a bounded function $f:\mathbb Z\rightarrow\mathbb R$ has a unique mean value if for every pair of finitely additive translation invariant ...
- 6,034
32
votes
1
answer
4k
views
Do invariant measures maximize the integral?
Update: The negative answer to the following question has been provided by Matthew Daws, who won, but also rejected, the bounty of 100 euro that I set over the question.
Let $\mathcal M(\mathbb Z)$ ...
- 6,034
15
votes
3
answers
3k
views
Entropy of a measure
Let $\mu$ be a probability measure on a set of $n$ elements and let $p_i$ be the measure of the $i$-th element. Its Shannon entropy is defined by
$$
E(\mu)=-\sum_{i=1}^np_i\log(p_i)
$$
with the ...
- 6,034
11
votes
4
answers
2k
views
Are measurable automorphism of a locally compact group topological automorphisms?
Consider a locally compact group $G$, considered as a measurable space with the completed Borelstructure wrt. the Haarmeasure. Consider a map $f:G \to G$, which is measurable and has an inverse, which ...
- 11.2k
3
votes
2
answers
2k
views
If $G$ is amenable, when $G\times G$ is amenable ?
I am not specialist on Topological Group Theory, I apologize if this is a trivial question.
Question. If $G_1=G_2$ are amenable topological groups what additional hypothesis we have to consider on ...
- 2,044
7
votes
5
answers
3k
views
Amenable exponential growth
Dear forum members,
Does anyone have a clear example of an amenable group with exponential growth?
Is real that if G is virtually amenable (has an amenable subgroup of finite index) then it is ...
- 375
14
votes
7
answers
3k
views
Cheap, non-constructive, free group generating rotations for Banach-Tarski
Stan Wagon's exposition of Banach-Tarski (for example) includes a beautiful explicit construction of two 2-sphere rotations which generate a free subgroup of the rotation group.
For teaching purposes ...
- 17.6k