Everything the deals with properties and definitions of Haar measure, as well as related fields when the question relies heavily on the notion of haar measure - group harmonic analysis, group ergodic theory etc.

learn more… | top users | synonyms

2
votes
0answers
100 views

Haar Functionals and Coquasi-triangular Structures

In this question it is mentioned that the coordinate algebra $C_q[G]$ Drinfeld--Jimbo algebras, for $G$ a compact semi-simple Lie group, admit a unique positive definite Haar functional. I was ...
2
votes
1answer
489 views

Measures and structure on conjugacy classes

Given a locally compact group $G$, does there exist a measure $\nu$ on the conjugacy classes $conj(G)$ such that for $f \in C_c(G)$ $$ \int_G f(g) d \mu_G(g) = \int_{conj(H)} \int_{G / G_\gamma} f(g\...
7
votes
2answers
1k views

Haar measure for large locally compact groups

In this answer, Gerald Edgar mentions that Haar measure is naturally defined on the $\sigma$-algebra of Baire sets (the smallest $\sigma$-algebra that contains all the compact $G_\delta$ sets) of a ...
16
votes
2answers
757 views

Haar measures in Solovay's model

Haar measure is a measure on locally compact abelian groups which is invariants to translations. For example, the Lebesgue measure on the reals is such measure. It can be shown without the use of the ...
13
votes
2answers
2k views

Volume of fundamental domain and Haar measure

In my research, I do need to know the Haar measure. I have spent some time on this subject, understanding theoretical part of the Haar measure, i.e existence and uniqueness, Haar measure on quotient. ...
2
votes
1answer
988 views

Haar measure of a subgroup

What is the connection between the normalized Haar measure of a compact group and the normalized Haar measure of one of its compact subgroups? I am trying to solve the following problem: Given $G$ ...
1
vote
0answers
201 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 ...
24
votes
4answers
6k views

When is $L^2(X)$ separable?

I have never studied any measure theory, so apologise in advance, if my question is easy: Let $X$ be a measure space. How can I decide whether $L^2(X)$ is separable? In reality, I am interested in ...
4
votes
2answers
1k views

How to define the quotient of a measure which is invariant under group action?

I am a physicist, and I have the following problem. Consider a locally compact group G acting over a measure space $(X, {\cal B}, \mu)$, and assume that $\mu$ is G-invariant. My problem is how to "...
21
votes
3answers
976 views

In an inductive family of groups, does the probability that a particular word is satisfied converge?

We have some group word $w$ in $k$ letters. We say a $k$-tuple of group elements $\vec{g} = (g_1, g_2, \ldots , g_k) \in G^k$ satisfies the word $w$ if $w$ gives the identity at $\vec{g}$. More ...
22
votes
4answers
1k views

Is every probability space a factor space of the Haar Measure on some group?

Let P be an arbitrary probability space. I would like to find a compact topological group $G$ so that the Haar probability measure on $G$ admits a measurable map to the probability space $P$. By a ...
31
votes
14answers
6k views

Explicit computations using the Haar measure

This question is somewhat related to my previous one on Grassmanians. The few times I've encountered the Haar measure in the course of my mathematical education, it's always been used in a very ...
3
votes
4answers
1k views

Measure on real Grassmannians

OK, so I'm reading about this nice measure you can define on a (real) Grassmannian on Wikipedia. Basically, and to save you the trip through the link, consider the Haar measure $\theta$ on $O(n)$, fix ...
18
votes
2answers
1k views

The Riemann zeta function and Haar measure on the profinite integers

In an answer to a question on MU about the Riemann zeta function, I sketched a proof that the probability distribution on $\mathbb{N}$ which assigns $n$ the probability $$\frac{ \frac{1}{n^s} }{\zeta(...
5
votes
3answers
1k views

Haar Measure Existence/A problem with Borel sets and regularity.

In Paul Halmos' Measure Theory book, section 53, he defines a content on a locally compact Hausdorff space to be a set function, $\lambda$ that is additive, subadditive, monotone, and $0\le\lambda(C)&...
6
votes
1answer
271 views

Measurable subgroups.

Let $G$ be a compact connected topological group and let $H$ be a subgroup of $G$. Suppose that $H$ is measurable with respect to the normalised Haar measure $\mu$ on $G$. Do we necessarily have $\mu(...
2
votes
1answer
551 views

Must a locally compact group be Hausdorff in order to possess a Haar measure?

Does the existence of (left) Haar measure on a locally compact topological group require that the group be Hausdorff?
5
votes
4answers
4k views

Haar Measure on a Quotient [closed]

Suppose you have a locally compact group G with a discrete subgroup H. Of course G has a unique (up to scalar) Haar measure, but it seems that G/H has and induced Haar measure as well. How does ...
4
votes
0answers
485 views

Haar measure on strictly upper triangular matrices

Let F be a function field, and A its adele ring. I want to consider U(A)/U(F), where U(A) is the space of strictly upper triangular matrices with entries from A, and U(F) is the same with entries ...
13
votes
8answers
3k views

Haar measure on a quotient, References for.

I remember reading Weil's "Basic Number Theory" and giving up after a while. Now I find myself thinking of it(thanks to some comments by Ben Linowitz). Right from the very beginning, Weil uses the ...