The haar-measure tag has no wiki summary.

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**14**answers

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### 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 ...

**21**

votes

**3**answers

935 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 ...

**21**

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**0**answers

934 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 ...

**20**

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**3**answers

4k 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 ...

**19**

votes

**4**answers

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 ...

**17**

votes

**2**answers

995 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} ...

**16**

votes

**2**answers

693 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

**7**answers

2k 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 ...

**10**

votes

**2**answers

1k 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. ...

**9**

votes

**3**answers

541 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 ...

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**2**answers

294 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 ...

**9**

votes

**1**answer

218 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
...

**9**

votes

**0**answers

131 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 ...

**7**

votes

**2**answers

551 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 ...

**6**

votes

**1**answer

260 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 ...

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votes

**2**answers

791 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 ...

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votes

**4**answers

3k 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 ...

**5**

votes

**1**answer

863 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 ...

**5**

votes

**1**answer

196 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 ...

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votes

**3**answers

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 ...

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**2**answers

906 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 ...

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votes

**1**answer

154 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

**1**answer

189 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 ...

**4**

votes

**1**answer

367 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 ...

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**0**answers

425 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 ...

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**4**answers

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 ...

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votes

**2**answers

197 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
...

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**1**answer

824 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 $G$ ...

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**4**answers

2k 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 ...

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**3**answers

215 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) ...

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votes

**1**answer

324 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 ...

**2**

votes

**2**answers

359 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 ...

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**1**answer

482 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?

**2**

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**1**answer

444 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} ...

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53 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
...

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79 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 ...

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99 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 ...

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304 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 ...

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113 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 ...

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vote

**1**answer

254 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 ...

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vote

**1**answer

92 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 ...

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39 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 ...

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144 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 $X$, ...

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98 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 ...

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186 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 ...

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**1**answer

378 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 ...

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**1**answer

228 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} ...

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**1**answer

436 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 ...