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.
170
questions
2
votes
0answers
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 ...
2
votes
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. ...
1
vote
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 $...
2
votes
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} ...
0
votes
0answers
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-...
2
votes
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 ...
2
votes
0answers
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,\...
0
votes
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 ...
1
vote
0answers
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 ...
1
vote
0answers
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 ...
1
vote
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 ...
0
votes
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 ...
1
vote
0answers
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 ...
1
vote
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 ...
3
votes
2answers
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}^...
1
vote
0answers
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 ...
2
votes
0answers
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 ...
0
votes
0answers
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 ...
2
votes
0answers
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(...
1
vote
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 ...
0
votes
0answers
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 itās Abelian. And there is a result says for a connected Lie ...
2
votes
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 ...
1
vote
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 $U(1)$,
$$M = [e^{i\...
1
vote
1answer
48 views
An application of Fubini's theorem I don't understand
Here is a result asserting that, under some technical conditions, we can replace a Borel almost everywhere homomorphism by a Borel homomorphism.
It is taken from Zimmer's book "Ergodic Theory and ...
2
votes
0answers
29 views
How does one invert a p-adic mellin transform, and what does it say about function asymptotics?
In ordinary analysis, given a sufficiently nice $f:\left[0,\infty\right)\rightarrow\mathbb{C}$, if we can compute the Mellin transform: $$\mathscr{M}\left\{ f\right\} \left(s\right)=\int_{0}^{\infty}x^...
1
vote
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 ...
1
vote
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 ...
2
votes
0answers
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 ...
0
votes
0answers
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 ...
1
vote
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 ...
4
votes
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 ...
1
vote
0answers
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 ...
-1
votes
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 ...
1
vote
0answers
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 ...
1
vote
0answers
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 ...
6
votes
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 ...
1
vote
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 $\...
1
vote
0answers
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 $v_i \in V$:
$$\int_G\langle \pi(g)v_1|v_2\...
1
vote
1answer
79 views
Haar measure over quotient spaces
In equation 23 of the following paper https://arxiv.org/pdf/1803.10743.pdf, the Haar integration over the quotient space $G/H$ is defined as follows
$$\int_{G / H} f(x) d x=\int_{G} f(g H) d g$$, ...
3
votes
0answers
42 views
Question about regular representation of compact group.
I first define the setting for my question. Let $G$ be a compact group with probability Haar measure $\mu_G$. Denote by $\lambda$ the left regular representation on $L^2(G)$ defined for $f \in L^2(G)$ ...
0
votes
0answers
26 views
Haar measure construction for extended Lie Algebra
Consider a Lie algebra $\mathcal{G}$ of the form $$[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, ...
2
votes
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, $G=(R_+,ā
)$. Since ...
0
votes
0answers
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 ...
2
votes
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-...
1
vote
0answers
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 ...
1
vote
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 ...
2
votes
0answers
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 ...
0
votes
0answers
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 ...
1
vote
0answers
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:...
1
vote
0answers
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 ...
