4
votes
1answer
151 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 ...
9
votes
3answers
565 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 ...
3
votes
1answer
173 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 ...
4
votes
1answer
355 views

Are there uniformly discrete paradoxical sets in $R^3$?

I think there aren't any discrete paradoxical sets in $R^2$ (any isometry that mapped a discrete set into itself would have to either be a glide-reflection, a translation or a rotation by $2\pi/n$, ...
15
votes
2answers
1k 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 ...
2
votes
1answer
329 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?
1
vote
1answer
244 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 ...
4
votes
1answer
242 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 ...
6
votes
4answers
579 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 ...
5
votes
0answers
172 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 ...
3
votes
0answers
192 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 ...
11
votes
3answers
619 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
1answer
320 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 ...
31
votes
1answer
3k views

100€ bounty ended: 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 $\mathcal M(\mathbb Z)$ ...
9
votes
3answers
923 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 ...
8
votes
4answers
766 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 ...
3
votes
2answers
819 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 ...
3
votes
4answers
824 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 ...
11
votes
7answers
2k 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 ...
3
votes
2answers
736 views

Hausdorff Dimension of Cayley Graphs of Groups

I was wondering what has been done concerning the Hausdorff measure of the Cayley graphs of finitely generated countable groups. There are number of issues that would need to be dealt with: 1.) By ...
36
votes
5answers
3k views

Why are abelian groups amenable?

A (discrete) group is amenable if it admits a finitely additive probability measure (on all its subsets), invariant under left translation. It is a basic fact that every abelian group is amenable. ...