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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Range (and other properties) of rotationally-invariant measure-producing mapping

Let $\Theta = \{\theta \in \mathbb{R}^d : \theta_1^2 + \cdots + \theta_d^2 \leq 1\}$ denote the unit ball in $\mathbb{R}^d$, and let $\mathcal{M}_1(\Theta)$ denote the set of probability measures on $\...
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Uniqueness of $\sigma$-finite left-invariant measure on measurable group

Perhaps this is a silly question with a "simple" counterexample, but I am not an expert in measure theory, so I thought of asking here. Basically, I am curious about uniqueness of $\sigma$-...
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Proof that every left $\chi$ covariant measure is a Radon measure

I am having a hard time understanding the proof found in Federer's "Geometric measure theory" that every left $\chi$ covariant measure over a locally compact Hausdorff groupe $G$ is a Radon ...
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How to see that a multiplicative Haar measurable function is "almost" a multiplicative character?

I have no idea about this problem (even with hint) in showing that a multiplicative Haar measurable function is equal to a multiplicative character almost everywhere. It is Exercise 11 in Tao's blog. ...
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Royal road to Abstract Harmonic Analysis?

I would like to learn Abstract Harmonic Analysis up to a level where research is feasible. My plan is the following: Read Loomis' book completely again. (I already read all the topics related to ...
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Is a random Wishart matrix $X \sim \mathcal{W}(I_d, n)$ equal in distribution to $U \Lambda U^T$ with $U$ uniform on $O(d)$ and $\Lambda$ chi-square?

My question is: Suppose that $X \sim \mathcal{W}(I_d, n)$ is it true that $X$ is equal in distribution to $U \Lambda U^T$ where: (i) $U$ and $\Lambda$ are independent, (ii) $U$ is drawn uniformly ...
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Is the Haar measure of the boundary of an open subset of the $k$-torus zero?

I have just recently started reading the basics of topological groups and Haar measures, and have become really curious about when the boundary of a non-empty open subset $U$ of a compact abelian ...
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Unique Haar measure on quotient is pushforward

Let $G$ be an abelian locally compact Hausdorff group with discrete subgroup $H$. Let $\mu$ be a Haar measure on $G$ and $\lambda$ the usual counting measure on $H$. Then we obtain a unique Haar ...
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Restriction of Haar Measure Induces a Restriction of Integral Formula

I just need some reality check. Suppose we have a locally compact abelian group $G$ with an open subgroup $H$. We equip $G$ with a Haar measure $\mu_G$, while we equip $H$ with the Borel measure $\...
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A subgroup of full measure is dense given a haar measure

I want to know why if $\mu$ is a haar measure on a compact $G$ and $\mu(A)=\mu(G)$ then $A$ is dense in $G$. This fact is mentioned in the wikipedia page, but I couldn't find a proof for it.
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How can we evaluate averaged products of random unitaries when the Weingarten function is singular?

In the random matrix theory literature, one often encounters identities associated with averages over ensembles of random unitaries. For a simple example let's say we're interested exclusively in $2\...
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A measure theoretic problem related to induced representations

I met such a rather concrete measure theoretic problem while dealing with induced representations of $p$-adic groups. So let $G$ be a unimodular locally compact group, with $P$ its closed subgroup (...
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Amenable groups, measure of volumes, and Paradoxical decomposition

I would like to kindly ask for your expertise on the following questions: It is known that the (additive) free commutative group Z^d is a discrete amenable group. Therefore, a finitely additive, left-...
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Haar measure on $PSL_2(\mathbb{R})$ via Iwasawa decomposition

I am missing something basic about the relation between the Haar measure on the group $G = KAN$ and the haar measures on the subgroups $K$, $A$, and $N$. Specifically, let $G=PSL_2(\mathbb{R})$ then ...
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The complement of a Zariski open dense subset in a real linear algebraic group has zero Haar measure?

Let $G$ be a real linear algebraic group (so it is locally compact and Hausdorff), equipped with a left-invariant Haar measure. Let $U$ be a Zariski open dense subset of $G$. I wonder how to show that ...
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Integrating an additive character over a local field

Let $F$ be a non-Archimedean local field, $\psi$ a non-trivial additive character of $F$. Let $\mathfrak{o}$ be the ring of integers of $F$, and $\mathfrak{p}$ be the maximal ideal of $F$. Endow $F$ ...
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Inner regular Haar measure

If $X$ is a locally compact Hausdorff space (l.c.H. space) then the Riesz representation theorem says that positive linear functionals on $C_c(X)$ are given by Borel measures on $X$. Unfortunately ...
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Let $G=SO(3)$ and $\theta:G\to G$ defined by $\theta(g)=JgJ$ where $J=\text{diag}(-1,1,1)$. Prove that, $\theta(g)\in Kg^{-1}K$ where $K=SO(2)$

Here $K=\left\{\begin{pmatrix}1 & 0\\ 0 & T_1\end{pmatrix}:\ T_1\in SO(2)\right\}$. $K$ is a compact subgroup of $G$. The above problem is part of the proof of $(G,K) $ is Gelfand pair. I've ...
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Radon-Nikodym derivative of Lebesgue measure with respect to Hausdorff measure is Jacobian term?

Consider the measurable space $(\mathbb{R}^n, \mathcal{B}(\mathbb{R}^n))$ with Lebesgue measure $\lambda$. I have a submanifold of $\mathbb{R}^n$ defined as the level set of a smooth function $f:\...
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Convolution on a locally compact group is associative

Consider the following fragment from Folland's book "A course in abstract harmonic analysis" (question is below the image). Can someone explain why the boxed equality is true? Don't we need ...
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Inherence of full measure sets from a Lie group to its homogeneous space.

Let $G$ be a Lie group with Haar measure $\mu$ and $\Gamma$ be a lattice of $G$. Namely $\Gamma$ is a closed discrete subgroup of $G$ with $G/\Gamma$ admitting a left $G$-invariant probability measure ...
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Estimating the distance from 1 to the conmutator of two elements in a Lie group.

I am reading Thurston's book (Three dimensional geometry and topology), and there is something I can't understand. Let $G$ be a Lie Group, in the book it says that there exists an $\epsilon > 0$ ...
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$G$ acts transitively on a space $X$. If a function on $X$ is $G$-invariant up to measure zero, is it necessarily a constant (up to measure zero)?

Consider a locally compact Hausdorff $σ$-compact topological space $X$ and a locally compact Hausdorff $σ$-compact topological group $G$ acting continuously and transitively on $X$ such that there ...
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2 votes
1 answer
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Question on integration on a local field

Let $F$ be a non-Archimedean local field, and $\mu_F$ a Haar measure on $F$. The space $C^{\infty}_c(F)$ of locally constant functions of compact support is spanned by characteristic functions of the ...
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How to write a Haar measure on $SO(n)$ and $SU(n)$ given Haar measure on $GL(n)$?

How to write a Haar measure on $SO(n)$ and $SU(n)$ given Haar measure on $GL(n)\,$? I know that $GL(n)$ has the Haar measure $(\det(A)^{-1} dA)$. I try to write from these a Haar measure for the ...
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$L^1$-density of the functions $F_{f,g}(s,t):=f(s^{-1}t)g(t)$ in $C_c(G\times G)\subset C_c(G,C_{0}(G))$

Suppose that $G$ is a locally compact group with Haar measure $\mu$. Note that we can view $C_c(G\times G)$ as a linear subspace of $C_c(G,C_{0}(G))$ by identifying the evaluations $F(s,t)$ and $F(s)(...
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Lemma 8.1 of "The Local Langlands Conjecture for GL(2)"

Let $F$ be a non-Archimedean local field, and $N$ the subgroup of $G=GL_2(F)$ of the form $\begin{pmatrix} 1 & x \\ 0 & 1 \end{pmatrix}$ with $x \in F$. Let $(\pi,V)$ be a smooth ...
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Haar integral, invariance and matrices

Wikipedia says: One property of a left Haar measure $\mu$ is that, letting $s$ be an element of $G$, the following is valid: $$\int_G f(sx) d\mu(x) = \int_G f(x) d\mu(x)$$ for any Haar integrable ...
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Crossing out an orthogonal-valued function out of expected value of product

Main part: Let $G$ be a locally compact group with corresponding Haar measure $\mu$. Let $f, g: \Omega \to G$ be arbitrary measurable functions. It would be really nice to have: $$\mathbb{E} [gf] = \...
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(Integrals of) continuous (integrable) functions $\mathbb{Z}_p\to\mathbb{C}$?

I'm looking for ways to compute some integrals of continuous (or integrable) functions $f:\mathbb{Z}_p\to\mathbb{C}$, where by $\mathbb{Z}_p$ I denote the $p$-adic numbers for a fixed prime $p$. ...
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4 votes
1 answer
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Volume of SU(2) and Haar Integral

The Pauli matrices are given by $$\sigma_1 = \begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix}, \quad \sigma_2 = \begin{pmatrix} 0 & -i \\ i & 0 \end{pmatrix}, \quad \sigma_3 = \begin{...
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Change of variable formula for Haar measure on product of Lie Groups

First let me recall the usual change of variable formula: Let $\varphi:\mathbb{R}^n\to\mathbb{R}^n$ be a bijection which is Frechet differentiable, $U\in\mathrm{Open}(\mathbb{R}^n)$, and $f:\mathbb{R}^...
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Integration over symmetric sets with respect to the haar measure

Let $G$ be a locally compact group and $m$ the corresponding Haar measure. Further let $S\subset G$ be symmetric $(S=S^{-1})$. Does then the following equation hold for $f \in L^1(G)$: $$\int\limits_{...
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What's the probability distribution for a single entry of a random unitary (or orthogonal) matrix?

Consider a random $n \times n$ unitary matrix, i.e. whose probability measure is the Haar measure for $\mathrm{U}(n)$. What is the marginal probability distribution for a single element of this matrix?...
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Modular function for certain upper triangular matrices (Anton Deitmar Automorphic Forms Exercise 3.4)

I am doing the following exercise from Anton Deitmar's Automorphic Forms Let $B$ be the group of all real matrices of the form $\begin{bmatrix}1 & x\\ 0 & y\end{bmatrix}$ with $y\neq 0$. Show ...
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2 votes
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Dirac Measure on a Group

In certain works https://arxiv.org/pdf/1803.11173.pdf (page 6, equation 8), the authors use the Dirac measure on the unitary group $\delta(V-U)dU$ to mean that "U must be of form V", where $...
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7 votes
3 answers
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Constructing the Haar measure of the $n$-dimensional Torus

Let $\mathbb{T}^n:=\mathbb{R}^n/\mathbb{Z}^n$ be the quotient of the group $(\mathbb{R}^n,+)$ by the subgroup $(\mathbb{Z}^n,+)$. I'm trying to construct the Haar measure of $\mathbb{T}^n$. I ...
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Haar measure of $O(n)$ vs. $U(n)$?

I am interested in the Haar measure on $U(n)$ resp. $O(n)$. I had the idea that you cannot "restrict" the Haar measure on $U(n)$ to Borel sets on $O(n)$, otherwise you would get a zero ...
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3 votes
1 answer
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The domain of the Haar measure of the $n$-dimensional Torus

I'm trying to find the Haar measure of the $n$-dimensional Torus. To find this measure I'm using the notion of pushforward measure of the Lebesgue measure. To understand my question please consider ...
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1 answer
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Connection between Haar measures on group and subgroup

Is there in general a connection between the Haar measure of a topological group and that one of its closed subgroup (with the induced topology)? I know that for $O(n)$ with the induced standard ...
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3 votes
1 answer
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If Haar measure is $\sigma$-finite, is the underlying topological space $\sigma$-compact?

Let $X$ be a locally compact Hausdorff group and $\lambda$ a left Haar measure on $X$. Assume that $\lambda$ is $\sigma$-finite. Is it true that $X$ is $\sigma$-compact? Attempt: Write $X = \bigcup_n ...
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3 votes
2 answers
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Is Haar measure reflection invariant?

Let $G$ be a locally compact group and $\mu$ be a Haar measure on $G.$ Is $\mu$ necessarily reflection invariant i.e. can we always say that $\mu (E) = \mu (E^{-1})\ $? where $E \subseteq G$ is a ...
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Haar measure of SU(2): Align to the z-direction for class functions

Any element of $\mathrm{SU}(2)$ can be parametrized via $$g=e^{i\varphi\vec{n}\cdot\vec{\sigma}}$$ where $\vec{n}$ is a normal vector and where $\varphi\in [0,2\pi]$. The matrices $\vec{\sigma}=(\...
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3 votes
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Integrating $\delta(g^{2})$ over $\mathrm{SU}(2)$

I am trying to calculate the following integral: $$\int_{\mathrm{SU}(2)}\,\mathrm{d}g\,\delta(g^{2}),$$ where $\mathrm{d}g$ denotes the normalized Haar measure on $\mathrm{SU}(2)$ and where $\delta(g)$...
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  • 1,562
6 votes
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Proof of formula involving the Haar measure of SU(2).

I would like to verify that $$\int_{\mathrm{SU}(2)}\mathrm{d}g\,\delta(g)=1$$ where the "delta-function" is defined via $$\delta(g):=\sum_{j\in\mathbb{N}_{0}/2}(1+2j)\chi^{j}(g)$$ where $\...
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  • 1,562
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Is $Y$, obtained from a random uniform unitary, uniformly distributed?

Recently, I came across a statement about random unitary transformations, but in the given context it is not clear to me whether this is a general statement or holds just in special cases. My own ...
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1 answer
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Function defined by diagonalized L2-function of two variables is in L2?

Lets take a complex-valued function $f\in L^{2}(G^{2},\mathbb{C})$, where $G$ is some compact Lie group and where the $L^{2}$-space has to be understood with respect to the normalized Haar measure on $...
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  • 1,562
1 vote
1 answer
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Is a function obtained from some L2 function by fixing some of its variables again in L2?

Lets take a complex-valued function $f\in L^{2}(G^{d},\mathbb{C})$, where $G$ is some compact Lie group and where the $L^{2}$-space has to be understood with respect to the normalized Haar measure on $...
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  • 1,562
1 vote
2 answers
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A finite-volume problem in the adelic group of a global field

I am reading Zeta functions of simple algebras by Roger Godement and Hervé Jacquet. In p139-140 they introduce a version of the theory of reduction. To get the finiteness of the Siegel domain module ...
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2 votes
1 answer
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Haar measure and Lie group

So in my textbook i get this expression that i just quite dont understand. Let $G$ be a compact group, and $\mu$ the normalised Haar measure on G. Let $(\pi, \mathcal H)$ Be a unitary representation ...
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