# Tagged Questions

**4**

votes

**1**answer

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

**3**answers

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

**1**answer

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

**1**answer

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

**2**answers

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

**1**answer

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

**1**answer

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

**1**answer

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

**4**answers

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

**0**answers

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

**0**answers

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

**3**answers

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

**1**answer

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

**1**answer

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

**3**answers

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

**4**answers

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

**2**answers

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

**4**answers

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

**7**answers

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

**2**answers

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

**5**answers

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. ...