Questions tagged [haar-measure]

Use this tag for questions related to the Haar measure, which is an assignment of an "invariant volume" to subsets of locally compact topological groups, consequently defining an integral for functions on those groups.

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17 views

Haar Measures of $U(d)$ and $O(d)$

I'm trying to compute the Haar measures of finite unitary group $U(d)$ and finite orthogonal group $O(d)$. I've shown that both are compact groups. Most of computings I've seen base on Peter-Weyl ...
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1answer
31 views

Reference for $p$-adic Haar integral

I have stumbled upon the notion of a $K$-valued Haar integral on a locally compact group, where $K$ is a non-Archimedean field, as well as the $K$-valued modular function, in an article of Schikhof. ...
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1answer
38 views

Defining left Haar measure on locally compact group

Let $G$ is a locally compact group that is homeomorphic to an open subset (say $U$) of $\Bbb R^d$ ,and let $\varphi$ be a homeomorphism of $G$ onto $U$. Show that if for each $a$ in $G$ the function $...
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1answer
88 views

Mistake in proof that affine linear group is unimodular

Where is the mistake in the following argument? I am arguing that the affine group on $\mathbb{R}^n$ is unimodular, which it is not!. Let $$\operatorname{Aff}_n(\mathbb{R}) := \left\{\begin{pmatrix} ...
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15 views

Isometrically embedding $L^1(H)$ into $L^1(G)$

Let $G$ be a locally compact group and $H$ a closed subgroup of $G$. Denote $\mu_G$ as Haar measure on $G$ and $\mu_H$ as Haar measure on $H$. Is it possible to embed $L^1(H)$ into $L^1(G)$? By Radon-...
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1answer
19 views

left invariant and right invariant imply inversely invariant

This is an exercise on page 158 of Conway's text A course in functional analysis: Let $G$ be a locally compact group. If $m$ is a regular Borel measure on $G$, show that any two of the following ...
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26 views

What is the $p$-adic Haar measure of this set?

Let $f_i(x1,\ldots,x_m)$ be polynomials in $m$ variables over $\mathbb{Z}_p$, for $i=1,\ldots, r$. Let $N_n$ be the number of elements in $\{x\mod p^n\mid x\in\mathbb{Z}_p^m, f_i(x)=0\text{ for }i=1,\...
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1answer
25 views

Proof of the $p$-adic Haar measure of $\{w\in\mathbb{Z}_p\mid \lvert w\rvert\leq 1/p^n\}$

The Haar measure on the $p$-adic integers is a measure which is translation invariant, finite for compact sets and positive for sets with non-empty interior. As such we can define it so that the Haar ...
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33 views

Haar measure on unit sphere

I am reading a paper where weak solutions to the Euler equations should be found by using the concept of convex integration. Therefore the proofs are very short and I've got some problems ...
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26 views

Radon product of Haar measure is a Haar measure

Suppose we have a family of compact topological groups $\{G_\alpha\}_{\alpha \in A}$, define $G=\Pi G_\alpha$, it's a compact toplogical group equipped with coordinatewise multiplication and product ...
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1answer
70 views

Right translation is a continuous action for fixed $f\in L^2(G)$

Let $G$ be a compact Lie group, and consider the space $L^2(G)$ with the Haar measure. For $f\in L^2(G)$, denote $L_gf$ by the map which sends $x$ to $f(xg)$. I want to show that the map $$ g\mapsto ...
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1answer
14 views

Gaussian orthogonal ensemble and Haar measure

I have been struggling with a probably easy question but I cannot prove it, so any insights would be really helpful. If I have a random matrix $X \in GOE(N)$, namely from the Gaussian orthogonal ...
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15 views

Haar measure on 2 by 2 upper right matrices

Let $G$ be $ \left( \begin{matrix} a & b \\ 0 & 1 \end{matrix} \right)$, which is a group endowed with matrix multiplication (This has been already shown). Suppose that ...
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1answer
53 views

Geodesics in compact Lie groups are exponential maps

I'm confused what exactly is happening in the following proof of Lemma 4.3 (Lemma 1.5 and Exercise 4.5 are also attached as they are apparently used in this proof). In particular, what does the ...
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83 views

Independence of some events with respect to Haar measure

In order to fulfill details in a proof, I came across with the following fact which, whenever true, allows me to conclude the line of reasoning. Here is the stage: I have an abelian group $G=\Bbb{Q}^...
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53 views

Connection between Haars measure Hilbertspace and Quantum mechanics.

Iam a physics student,I don't have any background in measuring theory. What I know is group theory and some priliminary basics of topology. In a book by MV Altaisky I see I don't understand how he ...
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50 views

Riemannian volume vs Haar measure?

Consider $SL(d,\mathbb{R})$ or $GL(d,\mathbb{R})$. Is it true that the Riemannian volume measure $\mathrm{vol}$ on these Lie groups as submanifolds of $\mathbb{R}^{d \times d}$ is the same as the Haar ...
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7 views

Existence of set of positive Haar measure containing torus

I am struggling around with the following problem, and was hoping someone could help. Let $G$ be a non-abelian compact connected Lie group with Haar measure $\nu$. Fix $h\in G$ and consider the ...
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29 views

Showing a function is not ergodic

Suppose $F:X\to X$ is ergodic with probability measure $\mu$ and $G$ is a nonabelian compact connected Lie group with Haar measure $\nu$. Let $h\in G$, and define $f_h:X\times G\to X\times G$ by $f_h(...
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1answer
55 views

Haar measure is absolutely continuous with respect to Lebesgue measure?

Is the Haar measure on a real Lie group absolutely continuous with respect to a Lebesgue measure? I just got introduced to the Haar measure and Lie groups. I know measure theory, but my background in ...
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8 views

Is $O(2)$ unimodular? (i.e. it left Haar measure is also right invariant.)

Is $O(2)$ unimodular? (i.e. it left Haar measure is also right invariant.) We know its connected component $SO(2)$ is unimodular because it’s Abelian. And there is a result says for a connected Lie ...
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1answer
47 views

Group Convolution is Associative

Let $G$ be some locally compact group and $\mu$ its associated Haar measure. I am trying to adapt this proof that convolution on the locally compact group $(\Bbb{R},+)$ is associative. Here's what I ...
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2answers
25 views

Parametrization of a matrix drawn randomly from $SU(n)$ (using Haar measure)

I have been trying to find a (simple) parametrization of a random Unitary matrix, drawn from $SU(n)$, in terms of random variables. A trivial example would be a matrix drawn from $U(1)$, $$M = [e^{i\...
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1answer
48 views

An application of Fubini's theorem I don't understand

Here is a result asserting that, under some technical conditions, we can replace a Borel almost everywhere homomorphism by a Borel homomorphism. It is taken from Zimmer's book "Ergodic Theory and ...
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29 views

How does one invert a p-adic mellin transform, and what does it say about function asymptotics?

In ordinary analysis, given a sufficiently nice $f:\left[0,\infty\right)\rightarrow\mathbb{C}$, if we can compute the Mellin transform: $$\mathscr{M}\left\{ f\right\} \left(s\right)=\int_{0}^{\infty}x^...
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2answers
32 views

computing integral over units in p-adic integers

I saw an integral in a paper and tried computing something similar, can anyone confirm the answer? Let $\nu$ be the valuation on $\mathbb{Q}_p$, $\mu$ the Haar measure on $\mathbb{Z}_p^{\times}$ and ...
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1answer
33 views

Haar measure on $p$-adic unit circle

I am trying to wrap my head around the haar measure on the unit circle in $\mathbb{Q}_p$ for $p< \infty $, that is the haar measure in $T_p = \{x \in \mathbb{Q}_p: |x|_p =1 \}$. Can anyone refer me ...
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45 views

Is regularity needed for uniqueness of Haar measure on compact groups?

I've looking at the following proof of the uniqueness of Haar measure on compact groups. I think I understand every step in the proof but I don't see where regularity is used. I see that the both fact ...
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20 views

An example of a non-invariant measure on a compact Lie group

I would like to construct a non-invariant measure on a compact Lie group but I'm not sure what is allowed and what the consequences are. Take the simplest example of $SO(2)$. The unnormalized ...
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1answer
42 views

In what sense are these two invariant measures on $SU(2)$ proportional?

An element $g$ of $SU(2)$ is of the following form: $$ g=\begin{bmatrix} z_1 & z_2\\ -\bar{z}_2 & \bar{z}_1 \end{bmatrix}, $$ where $z_i$ are complex satisfying $|z_1|^2+|z_2|^2=1$. I can ...
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1answer
66 views

Haar measure compact group

Let $G$ be a compact group with Haar measure $m$, then $m$ is left-invariant, in the sense that $\int_{G} f(x) \ dm(x) = \int_{G} f(sx) \ dm(x)$ for all $s\in G$ and for all $f\in C(G)$, and $m$ is ...
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16 views

Doubt on Haar intergrals

Let $(C,G,\alpha)$ be a $C^*$ dynamical system where $G$ is a locally compact Hausdorff topological group. and let $C_c(G,A)$ be the collection of compactly supported continuous functions from $G \to ...
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1answer
53 views

Zeta-function of simple algebras over $\sharp$-fields

What is the Zeta-function $\zeta_A$ of simple algebra over ${\Bbb Q}$ when $A \colon= {\mathrm{M}}_2({\Bbb Q})$ $A \colon= \{{\Bbb Q} \oplus {\Bbb Q}x \oplus{\Bbb Q}y \oplus {\Bbb Q}xy\,|\,x^2 = y^2 ...
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36 views

Expectation/Variance of SO(n)

I'm working on the following problem: Let $G = SO(n)$ for the group of rotation matrices on $\mathbb{R^n}$ with $n \geq 2$. Consider the Haar measure $\mu $ on $G$, and let $e_i$ denote the standard ...
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33 views

Orthogonality of $L^2(G)$ for a compact group

I am having trouble with the following: Let $G$ be a compact group with Haar measure $\mu$ and $(\rho,\mathbb{C}^n)$ be an irreducible representation of $G$. Naturally, we can suppose that $\rho$ is ...
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1answer
138 views

Representation theory, orthogonality

I am trying to get my head around representation theory and was hoping for some help. I will write out some text and then ask questions and make comments after. Let $G$ be a compact connected Lie ...
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1answer
86 views

Pushforward of product measure

I am having some trouble with the following. Suppose $X$ is a bounded metric space with Borel probability measure $\mu_X$. Let us suppose there is another probability space $(Y,\mu_Y)$ such that $\...
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34 views

Identity for Haar measure on compact group

Let $(\pi, V)$ be an irreducible unitary representation of a compact group $G$ with Haar measure $dg$. I would like to prove the following identity for $v_i \in V$: $$\int_G\langle \pi(g)v_1|v_2\...
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1answer
79 views

Haar measure over quotient spaces

In equation 23 of the following paper https://arxiv.org/pdf/1803.10743.pdf, the Haar integration over the quotient space $G/H$ is defined as follows $$\int_{G / H} f(x) d x=\int_{G} f(g H) d g$$, ...
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42 views

Question about regular representation of compact group.

I first define the setting for my question. Let $G$ be a compact group with probability Haar measure $\mu_G$. Denote by $\lambda$ the left regular representation on $L^2(G)$ defined for $f \in L^2(G)$ ...
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26 views

Haar measure construction for extended Lie Algebra

Consider a Lie algebra $\mathcal{G}$ of the form $$[T_i, T_j] = f_{ij}^k T_k$$ which has an Abelian (corresponding to the Abelian Lie algebra $A$) central extension $\mathcal{H}$ of the form $$[T'_i, ...
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1answer
74 views

$\mathrm{d}\mu(x) = \mathrm{d}\lambda(x)/x$, $\lambda$ is Lebesgue. Is $L^{\infty}([0,1],\lambda) = L^{\infty}([0,1],\mu)$?

Definitions: Let $\lambda$ be the Lebesgue measure on $G$. Define Haar measure $\mu$ on $G$ by $\mathrm{d}\mu(x) := \frac{1}{x}\mathrm{d}\lambda(x)$. $G$ is multiplicative group, $G=(R_+,⋅)$. Since ...
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68 views

Haar measure from Cartan metric?

Consider the following $d$-dimensional Lie algebra, $$[X_i, X_j] = f_{ij}^k X_k$$ so that the Cartan metric is given by $$g_{ij} = f_{ia}^bf_{jb}^a$$ Now, what I wish to know is whether the Cartan ...
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1answer
194 views

Volume element of Haar measure on SO(3) with Euler angle parametrization

I have a real-valued function $f(\alpha, \beta, \gamma)$ which takes Euler angles $\alpha, \beta, \gamma$ as input, that I would like to average over the uniform distribution on orientations of 3-...
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142 views

Haar measure on SU(n)

I can't seem to find a formula for Haar integral $$ \int_{SU(n)} F(U) d \mu$$ The integration will probably be executed using some parametrization with $n^2 -1$ parameters as I believe $SU(n)$ is ...
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1answer
38 views

Compute haar measure of a subset of SU(n)

Let $$X_n=\{(a,b,c)\in (SU(n))^{3} : c=aba^{-1}b^{-1}\}\subset SU(n)$$ with $SU(n)$ munished with its standard metric (say, normalized so that the total volume is $1$). Is there a good method to ...
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52 views

The history of Haar measure

Haar measure is a well-known concept in measure theory. Many books are perfectly dedicated to present its existence and uniqueness such as measure theory for D. Cohn. I am looking for a good ...
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24 views

maximum of entropies w.r.t. given transformation attained by Haar-measure?

Let $\mu$ be a Borel probability measure on $\operatorname{SL}_3(\mathbb{R}) / \operatorname{SL}_3(\mathbb{Z})$. Also let $g \in \operatorname{SL}_3 (\mathbb{R})$ and $h_\mu (g)$ be the measure ...
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49 views

The absolute convergence of (say) $\sum\limits_{n \in \mathbb Z} f(x+n)dx$ when $f$ is continuous and in $L^1(\mathbb R)$

Here is a basic theorem from locally compact topological groups. Suppose $G$ is a locally compact abelian topological group with closed subgroup $H$, and Haar measures $dg$ and $dh$. Suppose that $f:...
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26 views

Haar-measure on Lie-Group has maximal entropy?

Let $G$ be a Lie group and $\Gamma \subset G$ a discrete subgroup, such that there is a left-invariant probability measure $\mu$ on $G / \Gamma$. Let $g \in G$. Is it true in general, that the entropy ...