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-3
votes
0answers
59 views

Connection between Haar measure of locally compact group G and Haar measure compact subgroup of it [on hold]

Is there a connection between the Haar measure of the locally compact group G and the Haar measure of a compact subgroup?
2
votes
0answers
32 views

Is the sumset of two Haar positive closed subsets of a Polish group non-meager?

A famous Steinhaus theorem says that if measurable subsets $A,B$ of a locally compact topological group $G$ have positive Haar measure, then the difference $AA^{-1}$ is a neighborhood of the unit and ...
2
votes
1answer
263 views

The Tensor product of algebra group

Let G is a locally compact group. Is the following true? The tensor product of $L^1(G)$ with $L^1(G)$ is $L^1(G \times G)$.
5
votes
1answer
116 views

Why is it possible to normalize the Haar measure on the quotient?

I just asked a question which is related to the one I'm about to ask, but I realized my question can be reduced to the following: let $G$ be a locally compact abelian group with Haar measure $\mu$, ...
0
votes
0answers
38 views

The use of Haar measure in the Blichfeldt-Minkowski Lemma

I'm trying to understand a proof of the following result Theorem: Let $K$ be a number field, and $|| \cdot ||$ the idelic norm (product of the normalized absolute values at each place). There ...
5
votes
1answer
153 views

explicit integrals over a Lie group

I am looking for families of invariant integrals $\int_G dg f(g)$ (where $dg$ is a Haar measure) over a semisimple Lie group that can be evaluated in closed form, together with references where I can ...
4
votes
0answers
89 views

Concentration of measure for uniform distribution on Stiefel manifolds

This is my first post on MO, so I hope the question is suitable. I am looking at the uniform distribution on the Stiefel manifold, but more specifically, at the uniform distribution on the ...
1
vote
1answer
76 views

Structure of locally compact non discrete topological division algebras without the use of Haar measure

There is a well-known structure theorem for locally compact non discrete topological division algebras, see here http://math.stackexchange.com/q/1160086/187521 (I repost it here because I think it ...
2
votes
1answer
195 views

An expectation of the product of random unitaries

I want to find the answer of $$\int dU \ U^m X \ U^{\dagger m}$$ Where $m\in\mathbb{N}$ and $U$'s are unitary matrices in $U(n)$ and $dU$ is a normalized Haar measure. $X$ is a given self-adjoint ...
5
votes
1answer
452 views

Some calculus in the orthogonal group $O(n)$

How can one compute each of the following matrices, explicitly: $$\int_{O(n)} e^{g}dg$$ or $$\int_{O(n)} g^{n}dg \;\;\;\;n\in \mathbb{N} \;\;n>1$$ What is the explicite entries of the resulting ...
4
votes
1answer
128 views

If $ F(x,\bullet) \in {L^{\infty}}(G,B) $ for all $ x \in G $, then is $ x \mapsto F(x,\bullet) $ strongly measurable?

Let $ (X,\Sigma,\mu) $ be a $ \sigma $-finite measure space and $ B $ a Banach space. A function $ f: X \to B $ is said to be strongly $ \mu $-measurable iff it is the almost-everywhere pointwise ...
10
votes
0answers
181 views

Status of the analog of the Haar measure on quantum groups

In (Masuda, Nakagami, Woronowicz)'s paper, in the introduction, the authors mentioned the deficiency common to both their and (Kusterman, Vaes)'s approach regarding the Haar state (or Haar measure ...
4
votes
1answer
216 views

When does convolution preserve the `size' of a function?

For a positive function $f$ and positive measures $\mu, \nu$, does $$\mu\ast f\leq \nu\ast f \Rightarrow \|\mu\|\leq \|\nu\|?$$ More details: Let $G$ be a locally compact group, $C(G)$ be the space ...
2
votes
0answers
162 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 ...
10
votes
1answer
281 views

Compact Quantum Groups and the Existence of the Classical Haar Measure

Before I state my question, let me provide the definition of a compact quantum group. Definition: An ordered pair $ \mathscr{G} = (\mathscr{A},\Phi) $ is called a compact quantum group if ...
1
vote
0answers
41 views

Differential of Haar integral function

Let $C^k$ action of a compact Lie group $G$ on $R^m$, $D(g)$ denote the differential of the map $x\in R^m \mapsto g(x) \in R^m$ at the origin and $\mu$ is the normalized Haar mesure on $G$, consider a ...
9
votes
1answer
181 views

Haar measurable sets and quotient maps

Let $G$ be a locally compact Hausdorff group with a Haar measure $\mu$, let $H$ be a closed normal subgroup of $G$, and let $q: G \to G/H$ be the quotient homomorphism. Let $\nu$ be a Haar measure ...
9
votes
2answers
363 views

Entropy for Haar measure on $O(n)$

Let $G$ be a locally compact group. A measure $\mu$ is the right-Haar measure on $G$ if for every $g\in G$ and $E\subseteq G$ Borel set $\mu(Eg)=\mu(E)$. It is known that every locally compact group ...
6
votes
1answer
179 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 ...
4
votes
0answers
118 views

The Haar integral on uniform spaces

Does anybody know what the current research status is on the topic of Haar measures on uniform spaces? I'm specifically interested in compact uniform spaces and its corresponding Haar probability. As ...
1
vote
0answers
185 views

Question on the Quotient Integral Formula

I have a very concrete question on the proof of the following (see below): Given a 'nice' top. space $X$ and a 'nice' group operation of $G$, say, from the right, on $X$ and a certain measure on $X$, ...
2
votes
2answers
227 views

Banach algebra for measures induced by Haar measures

It is classical that $L^1(G, m)$ is a Banach algebra when $G$ is a locally compact group with Haar measure $m$ by using the operation of convolution via the integral ...
10
votes
3answers
664 views

To what extent has the Haar measure been generalized?

It is known that all locally compact groups, and therefore compact groups, have a left-invariant Haar measure which is unique up to scalar constant, also a right-invariant one. Is there a strictly ...
5
votes
1answer
232 views

Haar measure on $O(n)$ reduced to simpler probability space

The background of this question is how a random variable $X$ on the orthogonal group $O(n)$ whose distribution is the normalized Haar measure $\mu$, i.e., $\mu( O(n) ) = 1$, can be realized on a ...
2
votes
0answers
128 views

Chen's iterated integrals and free loop space

I recently found out about Chen's iterated integrals for paths in a differentiable manifold, and I was wondering if an analogous construction exists for free loops, i.e. a set of variables one ...
2
votes
3answers
248 views

A question on Haar measure on local field.

Let $F$ be a local field of characteristic 0, and $f:F\rightarrow \mathbb{C}$ be an integrable function. Is the following formulation valid? $ \int_{F^\times}f(x^2) d^\times x=\int_{F^{\times 2}}f(x) ...
2
votes
2answers
471 views

How do these two Haar measures on SL(2,R) compare?

By using the Iwasawa decomposition, one obtains a (bi-invariant) Haar measure on $G:=\mathrm{SL}(2,\mathbb{R})$ which can be symbolically written as ...
1
vote
1answer
97 views

Is function from topological group to metric space Borel?

Let $G$ be a pseudometrizable compact abelian topological group, $X$ a compact metric space and $f:X\rightarrow G$ a continuous bijective function. Suppose there exists $g\in G$ such that if ...
1
vote
1answer
297 views

Mean value theorems for the Haar integral?

Let $G$ be a compact topological group (feel free to add hypotheses if necessary). Is there any mean value theorem for its (normalized to 1) Haar integral? In general, are there mean value theorems ...
2
votes
1answer
426 views

Is every subgroup of a connected unimodular (matrix) Lie group also unimodular?

My intuition is that the answer is yes: Let $G$ be the original group, and let $H$ be a subgroup of $G$. Let $\mu$ be a Haar measure on $G$ that is both right- and left-invariant. I think that if we ...
3
votes
0answers
377 views

Haar measure on Galois groups

Galois groups are nice compact Hausdorff groups, and therefore possess a bounded Haar measure, unique if we insist that the total volume be $1$. What is the Haar measure on the absolute Galois group ...
1
vote
1answer
237 views

Harmonic Analysis [closed]

Let ‎$‎G‎$ ‎be a‎ ‎locally ‎compact ‎group‎, ‎$‎H‎$ ‎be a‎ ‎closed ‎subgroup ‎and ‎‎$‎N‎$ ‎be a‎ ‎normal ‎subgroup ‎of ‎‎$‎G‎$ ‎such ‎that ‎‎$‎H‎\subseteq ‎N‎$‎. ‎How ‎can ‎we get $$\int_{G/H} ...
21
votes
0answers
1k views

Correspondence between eigenvalue distributions of random unitary and random orthogonal matrices

In the course of a physics problem (arXiv:1206.6687), I stumbled on a curious correspondence between the eigenvalue distributions of the matrix product $U\bar{U}$, with $U$ a random unitary matrix and ...
2
votes
0answers
130 views

Almost conjugation-invariant neighborhoods of units in locally compact groups

Let $G$ be a locally compact topological group with unity $e$ and left Haar measure $m$. Let also $g\in G$ be a given element and $U$ a neighborhood (of compact closure) of $e$. I am interested to ...
0
votes
1answer
420 views

Positive function with zero Haar integral

If $C$ is a compact (semi-)topological (semi-)group, are there nonzero positive functions having zero Haar integral? In other words: is the Hermitian product associated to the Haar integral ...
7
votes
2answers
640 views

Integration on the space of symmetric matrices

Let $\mu$ be a Lebesgue measure on the space $G$ of real symmetric $n \times n$ matrices (the Haar measure on the additive group of such matrices). For any $A \in G$ let $\chi_{A}(x)$ be its ...
0
votes
1answer
468 views

Difference between spaces of integrable functions w.r.t Lebesgue measure and Borel measure [closed]

Is there a difference between $L^p(\mathbb R,\mathfrak B,\beta)$ and $L^p(\mathbb R,\mathfrak L,\lambda)$ ? Here I denoted by $\lambda$ the Lebesgue measure, defined on the Lebesgue $\sigma$-algebra ...
4
votes
1answer
396 views

Decomposition of Haar measure other than Hurwitz's

Hurwitz defined a decomposition of the Haar measure on $SO(n)$ based on Given's rotation. So by left multiplication of Givens rotation one can always bring an orthogonal matrix into the identity. The ...
6
votes
2answers
1k views

Statistics for Haar measure of random matrices?

Let's say I have $M$ samples of $N\times N$ real orthogonal matrices. What statistics can I calculate to test if they could have been drawn from a distribution consistent with Haar measure over ...
6
votes
4answers
3k views

Intuition for Haar measure of random matrix

What is an intuitive way to understand Haar measure as defined for random matrices, say, $N\times N$ orthogonal or unitary matrices? My understanding for what Haar measure means for $U(1)$ is that it ...
2
votes
0answers
99 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
469 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} ...
6
votes
1answer
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
733 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 ...
10
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
923 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
197 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 ...
22
votes
3answers
5k 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
962 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 ...