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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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Modular function and left Haar measure

In what follows, $G$ is a locally compact group with left Haar measure $\lambda$ and modular function $\Delta$. I came across this statement : It follows from a careful application of Hölder's ...
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Haar measure of a Lie group is (locally) a Lebesgue measure times density?

Preliminaries: Let be $G\subset\operatorname{GL}(n,\mathbb R)$ a closed subgroup of the general linear group over $\mathbb R$. Especially, $G$ is a Lie group by the closed subgroup theorem and we set $...
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Simple Lie Algebra + $Z(G) = \{1\}$ $\Rightarrow$ Simple Lie Group.

Let $G$ be a connected Lie Group ($\mathrm{dim}G>1$) such that $Z(G) = \{1\}$ (center of $G$) and $\mathrm{Lie}(G) = \mathfrak g$ is a simple Lie Algebra. How do I prove that these conditions imply ...
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Integration over a compact group with respect to Haar measure

Let $G$ be a compact abelian group and $\hat G={\rm hom_{\rm continuous}}(G,\mathbb{C}^*)$. Let $\mu$ be the Haar measure on $G$. For $\chi\in\hat G$, show that $$\int_G\chi\cdot\mu=1\quad\text{if}\...
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Harmonic Analysis on a Finitely-Generated Matrix Group

Let $\mathbf{M}_{1},\ldots,\mathbf{M}_{N}$ be invertible $2\times2$ matrices with entries in $\mathbb{Q}$ of the form: $$\mathbf{M}_{n}=\left[\begin{array}{cc} a_{n} & b_{n}\\ 0 & c_{n} \end{...
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Question in p-adic integration (Igusa type)

I am trying to learn how to solve Igusa type local zeta function. Ex. $$\int_{\mathbb{Z}_{p}}||x^3,x^2y,y^2||d\mu(x,y)$$ A nice method I was recently introduced to was to substitute $x=a+px'$ and $y=...
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Haar-measure in KAK decomposition of $SL(2,\mathbb{R})$

I would like to know if there is any way to calculate the Haar-measure of the Lie group $ SL (2, \mathbb{R}) $ with respect to the $ KAK $-decomposition, where: for each $ g \in SL (2, \mathbb{R}) $, ...
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Embedding of $L_p$ spaces of Metrizable Measurable Group.

Let $G$ be a $\sigma$-locally compact and metrizable topological group equipped with the left Haar measure $\mu$. When can $L^p_{\mu}(G)$ be Bi-Lipschitzly embedded into a Lebesgue space on a ...
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solving a $p$-adic integration involving maximum function

I am always struggling when it comes to dealing with maximum functions. I am trying to find the solution to this integral $$\int_{\mathbb{Z}_p^3}||xy,xz,yz||_p^sd\mu(x,y,z),$$ where $||xy,xz,yz||_p^...
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Haar Measures and Embeddings of $\nu$-adic integers

Let $\nu\geq4$ be any composite integer, and let $d\in\left\{ 2,\ldots,\nu-1\right\}$ be any non-trivial divisor of $\nu$. Since $d\mid\nu$, note that any sequence $\left\{ \mathfrak{y}_{n}\right\} _{...
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problem in $p$-adic integration

I am working on the $p$-adic integration and I am trying to find how to integrate $$\int_{\mathbb{Z}_p^2}||x,y||_p^sd\mu (x,y),$$ where $d\mu$ is the haar measure and $||x,y||_p^s=\sup\{|x|_p^s,|y|_p^...
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Affine Change-of-Variables Formula in $p$-adic Haar measure integration.

Let $p$ be a prime number, let $V$ be an arbitrary non-empty subset of $\mathbb{Z}_{p}$, and let $d\mu$ be the Haar measure on $\mathbb{Z}_{p}$ subject to the normalization $\int_{\mathbb{Z}_{p}}d\mu=...
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Some notational question related to Haar measure (what do $d g^{-1}$ or $d (hg)$ mean?)

I am looking at some brief introduction to Haar measures and since I'm not understanding basic notion, I would greatly appreciate any clarification. Let $G$ be a locally compact group and say we ...
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Haar measure on a Global Field

Let $\Bbbk$ be a finite-degree field extension of $\mathbb{Q}$. I am trying to figure out the formula for the unique Haar measure $d\mu_{\Bbbk}$ on $\Bbbk$ normalized so that: $$\int_{\Bbbk}\mathbf{1}...
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why is the pushforward of Haar on $SU(2)$ by trace the semicircle measure?

This is something which has been bugging me since I keep getting the wrong answer: that the pushforward measure should be the "square of the semicircle" rather than the semicircle measure. My ...
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Best Power in a Probability Inequality

Let $f:S^{n-1}\rightarrow \mathbb{R}_{+}$ be a Lipschitz function. For $1\leq k\leq n$, define $f_k:G_{n,k}\rightarrow \mathbb{R}_{+}$ by $f(E)=\max_{x\in S^{n-1}\cap E}f(x)$. Let $\sigma_k$ denote ...
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Is it true that the Haar covering numbers $(f:\varphi)$ converges to $\int_Gf\operatorname{d}\mu$ for $\varphi\to\delta$?

Let $(G,\cdot,\tau)$ be a topological group whose topology is Hausdorff and locally compact and whose identity is $e.$ Denote by $\mathcal{B}_\tau$ the family of Borel subsets of $(G,\cdot,\tau)$, i....
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Normalization of Haar measure

The following was stated in my course on automorphic forms and I think there is at least missing an assumption. Let $G$ be a locally compact Hausdorff group and $H\subseteq G$ compact. Then we can ...
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Different(?) measures on sphere

Let us consider the real sphere $S^{2n-1}:=\{x=(x_1,\ldots,x_{2n})\in\Bbb R^{2n}\mid\lVert x\rVert=1\}$ and the complex sphere $S_\Bbb C^{n-1}:=\{z=(z_1,\ldots,z_n)\in\Bbb C^n\mid\lVert z\rVert=1\}$. ...
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Haar measure from axis-angle representation of $SO(3)$

My aim is to study some special representations of $SO(3)$ using characters, and to do this I need an explicit expression of the Haar measure on $SO(3)$. I've found some "versions" of the Haar measure ...
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A problem from Folland: constructing Haar measure from Lebesgue measure

The following is a result from A Course in Abstract Harmonic Analysis by G.B. Folland. It also appears as an exercise in the real analysis text by the same author. The context is Haar measure on a ...
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Weak-$*$ topology on algebraic dual

I was looking at Izzo, Alexander J., A functional analysis proof of the existence of Haar measure on locally compact Abelian groups, Proc. Am. Math. Soc. 115, No. 2, 581-583 (1992). ZBL0777.28006. ...
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Uniform measure on an affine variety

I'm trying to find an algorithm that uniformly samples elements in the following variety: $A_p=\{(\lambda_1,...,\lambda_n) \in [0,1]^n / \sum \lambda_i = 1, \, \sum \lambda_i^2 = p\}$ I think it can ...
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What is the product of two Haar distributed unitary matrices?

I guess a product of two Haar distributed unitary matrices is also a Haar distributed unitary matrix. Is there a proof?
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Haar measures are decomposable

The definition of decomposable measures is as follows: (Here $\mathcal{M}$ is a $\sigma$-algebra over $X$.) My question is part c) of the following exercise: I have managed to prove a) and b). For c)...
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Upper-triangular subgroup is not unimodular

Consider the group $GL_n(\mathbb{Q}_p)$ of $n \times n$ invertible matrices over the $p$-adic field $\mathbb{Q}_p$. My goal is to prove that the subgroup $P_0$ of upper triangular matrices is not ...
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Is the set of measurable functions measurable?

Let $M$ be the set of measurable functions in $\mathbb{R^R}$. Now, $\mathbb{R^R}$ has the borelian sigma-algebra associated with its product topology, which allows us to ask the following question : ...
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Drawing random subspaces from Grassmannian with uniform probability

Consider the Grassmannian manifold $G(M, N)$ of $M$-dimensional subspaces in $R^N$. I want to approximate (stochastically) an integral of the form $$ \int_{G(M, N)} f(v) \, dv, $$ where $f : G(M, N) \...
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Measure on Lie Algebra “induced” by Haar measure on U(n)

On the unitary group $U(n)$, the Haar measure provides a suitable notion of uniformity for many applications. The Lie Group $U(n)$ is generated by the Hermitian matrices, i.e., I can write any $A \in ...
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Why $\frac{dx}{|x|}$ is a Haar measure on $\mathbb{R}\setminus\{0\}$

This is in "A course in the abstract harmonic analysis by G.B. Follan on page 45" Why $\frac{dx}{|x|}$ is a Haar measure on multiplicative group $\mathbb{R}\setminus\{0\}$ I've started as ...
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Definition of Haar integral in Bushnell and Henniart

In Bushnell and Henniart's The Local Langland's Conjecture for GL(2) they define a right Haar integral on a locally profinite group $G$ as being a non-zero linear functional $$ I: C^{\infty}_{c}(G) \...
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Why does ergodicity fail?

Apparently the following is not ergodic, and a very sketchy argument is given, so I was hoping someone here would be able to explain/give a better argument. Suppose $G$ is a compact connected non-...
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Haar measure of abelian subgroup of non-abelian group

I am trying to prove non-ergodicity of a certain map, and it boils down to the following. Suppose we have a compact connected Lie group $G$ with Haar measure $\mu$, and suppose that $G$ is non-...
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Haar-measure on homogenous spaces

Consider a locally compact group $G$ and a compact, closed subgroup $H$. It is well known that we have a Haar measure $\mu$ on $G$ and can then construct a left-invariant measure on $G/H$, which, as ...
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$\sigma$ finiteness of locally compact groups with haar measure

To my understanding, the Haar-measure on a locally compact group $G$ is inner regular on all open subsets and compact subsets have finite measure. Why then would the group itself, being open, not be $\...
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Bound on the measure of a sum of sets

Let $G$ be a locally compact abelian group (I am mostly interested in $\mathbb{R}^d$) with Haar measure $\mu$ be the Haar measure. Consider a fixed compact set $K$ containing the identity. Is it true ...
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Haar Measures on Solenoids

Consider the polynomial ring $\mathbb{Z}\left[x\right]$ in the indeterminate x. Fix an integer $b\geq2$. Applying the evaluation map $x\mapsto \frac{1}{b}$ then gives us the “$b$-adic” rationals; ...
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Euler angles on $SO(n)$

I would like to express the following integral $\int_{SO(n)} f(g) dg$ in terms of the Euler angles $\theta_1,\dots, \theta_n$. In a pdf I found this formula: $$\int_{SO(n)} f(g) dg= \int_{0}^{\pi} \...
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The average effect of a self-adoint projection

First, here is a brief non-rigorous version of the question: Staring with two arbitrary rays (one-dimensional subspaces) in a finite-dimensional Hilbert space, what is the average effect of a ...
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How to prove that if $f*\chi_A=0$ a.e. for all $A$ of finite measure then $f=0$ a.e.

Let $G$ a locally compact abelian group and let $\mu_G$ an Haar measure of $G$. Suppose that $f\in L^1(\mu_G)$ is such that for all measurable $A$ of finite $\mu_G$-measure it happens that $f*\chi_A=0$...
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What is the measure of this set of matrices

I am interested in the following question about matrices: Let $U\in\mathbb R^{n\times n}$ be considered as a map from $(\mathbb R^n,||\cdot||)$, a normed space, to itself. We say that $U$ is good if ...
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What is $\int x^4 \, d\mu_p$ on the circle $\{ x^2 + y^2 = 1 \}$ respect to Haar measure on $\mathbb{Q}_p$?

I am trying to understand integration with respect to Haar measure. Here are the first two examples I can think of. Let $X$ be the variety corresponding to the circle: $$ X = \{ x^2 + y^2 - 1 = 0 \}$...
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How is convolving Haar measure with itself over subsets enough to define subgroups?

In this post, Terry Tao says that Gaussians are "a subgroup of $\Bbb R$". When you convolve Haar probability measure on a (compact) subgroup with itself, you get back the same measure, and this can ...
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Are the Lie groups $GL_n(\mathbb R)$ and $GL_n(\mathbb C)$ unimodular?

I want to know whether $GL_n(\mathbb R)$ is a unimodular Lie group, that is, whether it has a Haar measure that is both left and right invariant. What I've tried so far is seeing $GL_n(\mathbb R)$ ...
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Existence of isometry-invariant measures

One of the wonderful things about the development of measure theory is that the most important theorems (Littlewood's principles, Dominated convergence, Fubini's theorem, etc.) can be proven in the ...
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Orthogonal group representation induces an isometry on $L^p$ spaces

I am reading a paper and I am unsure about the following. I will first explain the setup. Suppose we have a measure space $(X,B,\mu)$. Let $G$ be a compact Lie group with Haar measure $\nu$. Suppose ...
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Sufficient condition for Radon measure be Haar measure

I am reading Dinakar Ramakrishnan&Robert J.Valenza Fourier Analysis on Number Fields. I want to understand following proposition. Prop 1-7 (ii) Let $G$ be a locally compact group with ...
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Haar measure of $p$-adic integers with valuation 0 modulo 3

I am reading (trying to read) this paper, when I came across some statement regarding the Haar measure of $p$-adic integers (it appears in the proof of Lemma 2.5). The statement is the following: $$ ...
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Does Haar measure have such property?

Suppose, $G$ is a locally compact topological group. $\mu$ is Haar measure on that group. $A$ and $B$ are Borel subsets of $G$, such that $\mu(A)$ and $\mu(B)$ are finite. Does the inequality $\mu(\{...
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Finite measure fundamental domain for a discrete group implies it's a lattice

Here $G$ is a locally compact second countable topological group with left haar measure $\mu$, and $\varGamma$ is a discrete subgroup with a borel subset $\varOmega \subseteq G$ s.t. $G=\biguplus_{\...