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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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Running into strange conclusions when using the modular character to define a right Haar measure

In the following, $G$ is a locally compact, $\sigma$-compact metrizable group, and $m$ a left Haar measure. Exercise 10.5 in Einseidler & Ward's functional analysis book asks us to show that: $$m^{...
While I Am's user avatar
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Ex. 8.21 in Einseidler & Ward FA book: where do we need that $G$ is abelian?

I am working through the following exercise in Einsiedler & Ward's book Functional Analysis, Spectral Theory, and Applications. Exercise 8.21: Let $G$ be a compact metric abelian group. Show that ...
While I Am's user avatar
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Free probability version of Poincaré Separation Theorem

Suppose $A$ is a $d \times d$ real positive semi-definite matrix, and $U$ is a $d \times n$ semi-orthogonal matrix such that $U^\top U = I_n$. Define $B = U^\top A U$. The Poincaré Separation Theorem ...
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Calculating trace of inverse of a random matrix

I am trying to calculate $\frac{1}{r}$Tr$((\mathbf{I}_r-B)^{-1})$ where $\mathbf{I}_r$ is the identity matrix and B is a random $r$ by $r$ matrix given by $B = O^{T} DO$, where $O_{n\times r}$ is ...
Aaradhya Pandey's user avatar
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Is the Haar Measure on a Group the Only Reasonable Way to Define Randomness?

Given a compact topological group $G$ and a closed subgroup $H$. Let $d \mu_H$ and $d \mu_G$ be the respective unique Haar measures. In general, when we say "pick a random element of $H$," ...
a.e's user avatar
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The Haar measure on a product of a positive-measure set and an unbounded subgroup

Let $G$ be a locally compact group with left Haar measure $\lambda$, $A\subseteq G$ with $0<\lambda(A)$ and $H\leq G$ a closed and non-compact subgroup. Must $\lambda(A\cdot H)=\infty$? It is not ...
Uri George Peterzil's user avatar
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A p-adic integration involving additive character.

I was studying p-adic integral and I came across an example which is : \begin{equation} \int_{p^nZ_p^x} e_p(x) dx = \begin{cases} p^{-n}(1-p^{-1}),\text{if $n\geq0$}\\ -1, \text{if $n=-1$}\\ 0, \text{...
Soumyadeep mandal's user avatar
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Notation of Haar measure

Quick question about the notation of Haar measures: Consider the (multiplicative) group $G = \{ \begin{pmatrix} y & x \\ 0 & 1 \end{pmatrix} |\ x,y\in \mathbb{R}, \ y>0\}$. I read that the ...
luc.1401's user avatar
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A trick to prove existence of Haar measure

Let $G$ be an abelian compact and separable group, then there exists a unique Radon measure $\mu$ such that $\mu(g A) = \mu(A)$ for each $g \in G$ and Borel set $A$ $\mu(G) = 1$ The proof of this ...
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Haar measure and convolution on a Lie subgroup

Let $G$ be a Lie group and $H \leq G$ a closed Lie subgroup. Take Haar measures $dg$ and $dh$ on $G$ and $H$. Can we get $dh$ from $dg$ (up to some constant)? I would also like to understand it in ...
mixotrov's user avatar
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Lattices of Lie Groupoids

There exists an important concept in Lie group theory being that of lattice. Let $G$ be a Lie group, a lattice $\Gamma$ is a discrete subgroup $\Gamma \subseteq G$ such that the quotient $G/\Gamma$ ...
Tomás Pacheco's user avatar
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For a locally compact group that is not unimodular, find a Borel set that is finite for a left Haar measure and infinite for a right Haar measure

I am stuck with a fairly easy exercice: Let $G$ be a locally compact topological group that is not unimodular with left Haar measure $\mu$ on $G$. Show that there exists a Borel set $A$ such that $\mu(...
Picollus's user avatar
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$\mathrm{GL}_n(\mathbb Q_p)$ is unimodular

I'd like to prove the well-known result that $G = \mathrm{GL}_n(\mathbb Q_p)$ is unimodular, using elementary results, i.e. without reductive groups. Some definitions: as a locally compact group, $G$ ...
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An exercise on Haar measures in Lang's Real and Functional Analysis book

I'm trying to understand the statement of the following exercise (Lang, Real and Functional Analysis, page 326): Identify $\mathbb{C}$ with $\mathbb{R}^2$. Let $\mu$ be the Lebesgue (Haar) measure on ...
Matheus Frota's user avatar
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Equality of $\int_{S^{n-1}} \int_{S^{n-2}(\theta^\perp)} f(\theta,u) du d\theta = \int_{S^{n-1}} \int_{S^{n-2}(u^\perp)} f(\theta,u) d\theta du$

For a project I do, I need to use the following proposition in my calculation: Let $f: D \to \mathbb{R}$ be a non negative measurable function, where $D=\{(u,\theta) \in (S^{n-1})^2;u\perp \theta\}$, ...
Lord-Schnitzel's user avatar
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Computing Haar measure of matrices sampled from SO(n)

I am looking to sample uniform matrices from SO(n). I know that uniform matrices can be sampled from O(n) by taking the QR decomposition of Gaussian random square matrices and adjusting the sign of ...
magnesium's user avatar
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2 answers
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Haar Measure of Product of Compact Sets

This is from Bachman's Harmonic Analysis book, exercise 8.3 Let $\mu$ be a Haar measure in $G$ and let $E_1$ and $E_2$ be two compact subsets of $G$ such that $\mu(E_1)=\mu(E_2)=0$. Does this imply ...
Vinay Deshpande's user avatar
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Universal concentration of random matrix

I am studying a family of random matrices and observe a universal concentration behavior which I would like to understand. Let $U$ be an $N \times N$ random unitary drawn from the Haar measure, and $Z ...
nervxxx's user avatar
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Conditional distribution of random orthogonal projection matrix

I have encountered a rather curious question. Suppose I have a symmetric idempotent orthogonal projection matrix $A\in\mathbb R^{N\times N}$ that projects onto a uniformly random $n-$dimensional ...
Landon Carter's user avatar
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Definition of an integral over linear maps

Let $G$ be a locally compact Hausdorff topological group with Haar measure $\mu$. Let $\varphi \colon G \to GL(V)$ be a continuous representation of $G$, where $V$ is a Hilbert space over $\mathbb C$. ...
stupid_questions's user avatar
2 votes
1 answer
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Relating integration of forms to Haar measure integration on a Lie group

Let $G$ be a Lie group with Haar measure $\mu$ and Lie algebra $\mathfrak{g}$ given as the space of left-invariant vector fields on $G$. I want to understand the relationship between integration of ...
user920957's user avatar
3 votes
1 answer
102 views

Covering balls by balls of half radius in compact metric space (or, comparing Haar measure between balls of comparable radius)

Let $(X,d)$ be a compact metric space. Given $r>0$ I can obviously cover any ball $B(x,r)$ by finitely many balls $B(x_j,r/2)$, $j=1,...,k(r)$, by compactness. My question is whether there is ...
User's user avatar
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6 votes
2 answers
190 views

Calculating the Haar integral on $SU(2)$ in practice

I'm trying to calculate the Haar integral on $SU(2)$ of a given function $f: SU(2) \to \mathbb{R}$. For this particular function, I know the value of it on the subgroup $$T = \left\{ \begin{pmatrix} z ...
Robin's user avatar
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Haar measure and push forward in the subset of squares

Let $K$ be a compact group with Haar probability measure $m$, let $K^2 = \{k^2 : k\in K\}$ and suppose that $m(K^2)>0$. Show that if $m(A\cap K^2) = m(K^2)$ then $m(\{k\in K : k^2\in A\})=1$.
kenzie017's user avatar
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1 answer
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Absolute continuity of the push forward of the Haar measure

Let $K$ be a compact group with Haar measure $m$. Let $K^2 = \{k^2 : k\in K\}$ and suppose that $K = K^2 \cup wK^2$ for some $w\in K$. Let $\mu$ be the push-forward of $m$ under the map $k\mapsto k^2$....
kenzie017's user avatar
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Integration over the general linear group over $\mathbb Q_p$

Before asking, I apologize if this question is duplicated. It seems a sort of basic stuff and I tried to find the reference, but I couldn't find it(only the statement without the proof appears in the ...
LWW's user avatar
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$L^1(G)$ is a Banach *-Algebra [duplicate]

Let G be a locally compact group with left Haar measure $\mu$. In Principles of Harmonic Analysis, it's affirmed that $L^1(G)$ is a Banach *-Algebra, where the multiplication operation is convolution ...
Pedro Lourenço's user avatar
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Computing Integral over the Orthogonal Group $O(d)$

Question: I am seeking methods or references to numerically compute integral over the orthogonal group $O(d)$. The specific context is to compute integrals of the form: $$ \int_{O(d)} f(g) dg $$ where ...
Eddie Lin's user avatar
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2 votes
1 answer
138 views

Understanding definition of Adelic integral and calculate simple example

I'm trying to understand the definition the adelic integral given in Goldfeld and Hundley. It says that: Suppose that $f =\prod_v f_v$ is a factorizable function, that $f_\infty$ is an integrable ...
slowpoke's user avatar
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0 answers
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Haar measure on the positive real numbers

Consider the locally compact Hausdorff topological group $\mathbb{R}_{>0}$, I can use Haar theorem to show that there is a Haar measure on my borel sets. However, I am having trouble determining it....
3j iwiojr3's user avatar
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Volume of rank 1 matrices with bounded frobenius norm (to determine the covering number)

I am trying to bound the covering number of the set $M$ of rank 1 matrices (in $\mathbb R^{n\times d}$) with frobenius norm at most 1. I can do this with the following method: given a $\delta$-cover $...
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Quotient Integral Formula involving compact subgroup $K$ and closed subgroup $H$

Question: I am trying to follow the proof the following theorem from page 21 of Deitmar et al's Principles of Harmonic Analysis Let $G$ be a locally compact group, $K \subset G$ a compact subgroup ...
L-JS's user avatar
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3 votes
1 answer
179 views

Decomposing the unitary Haar measure as product of unit vector Haar measures

Let $\mu_{D}(U)$ be the Haar measure on the D-dimensional Unitary group $U(D)$, where $U \in \mathrm{SU}(D)$ or $U(D)$. Can we think of this measure as picking first a unit vector according to the ...
Soham Ghosh's user avatar
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Understanding of $d_{\mathrm{a}}X$ denoted as the Lebesgue measure on $\mathrm{Mat}_n(\mathbb{R})$.

In Exercise 1.1 of Lie Groups written by Daniel Bump, let $d_{\mathrm{a}}X$ denote the Lebesgue measure on $\mathrm{Mat}_n(\mathbb{R})$. I feel confused to understand the definition and properties of ...
一団和気's user avatar
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36 views

λ f ↦ ʃ f(z) dz/z is a Haar integral, but not λ f ↦ ʃ f(z)

I am confused about the uniqueness of Haar integrals. A Haar integral is a Radon integral determined by the corresponding Haar measure. Mostly I am interested in the following quite special integral, ...
user avatar
3 votes
1 answer
75 views

Pontrjagin duality for a topological ring

Let $R$ be a locally compact topological ring, and let $S$ be its Pontrjagin dual under addition. For instance, ℤ/nℤ is Pontrjagin dual to the ring ℤ/nℤ $\mathbb{Q}$/ℤ is Pontrjagin dual to the ring $...
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1 vote
0 answers
43 views

Ergodic Measures are Push Forwards of Haar Measure

Here is a problem I am trying to solve. Let $G$ be a compact group acting continuously on a locally compact metrix space $X$. Suppose $\mu$ is an $G$- invariant ergodic Borel probability measure of $...
JNF's user avatar
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Convolution of Double Coset Indicator Function in Hecke Algebra of Locally Profinite Group

Let $G$ be a locally profinite group (i.e. a topological group that is locally compact Hausdorff and totally disconnected or, equivalently, a Hausdorff topological group s.t. $1 \in G$ has a ...
Tragomix's user avatar
1 vote
1 answer
138 views

Integration of periodic functions on the real line with respect to the Haar measure of 1-dim. torus

A function $f:\mathbb R \to \mathbb C$ with period 1 can be identitied with a function defined on the 1-dimensional torus $\mathbb T = \mathbb R / \mathbb Z$, the latter being continuous if and only ...
Ulysse Keller's user avatar
1 vote
0 answers
27 views

Why is the $\text{weak}^*$-closedness important when considerning convex combinations of invariant Borel probability measures?

I got a question while reading page 148 of Katok and Hasselblat. Suppose $\mu$ is an $L_{g_0}$-invaraint Borel probability measure for the translation $L_{g_0}(h) = g_0h$ on a compact metrizable ...
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1 vote
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Generating Random Linear Subspace

Let $n$ and $m$ be two integers such that $m\leq n$. Let $G_{n,m}$ be the set of all m-dimensional linear subspaces of $\mathbb{R}^n$. Assume we want to generate a subspace of dimension $m$ which is ...
MMH's user avatar
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1 vote
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Explicit expressions for the integrals of second degree polynomials in the orthogonal group

Wikipedia has explicit formulas for first and second degree polynomial integrals in the unitary group using Weingarten functions (https://en.wikipedia.org/wiki/Weingarten_function). Is there any ...
Alexandru Meterez's user avatar
2 votes
1 answer
74 views

How often is a monic polynomial highly divisible by p?

Let $p$ be a prime, and let $|\cdot|_p$ be the $p$-adic absolute value. Let $f(x) \in \mathbb{Z}[x]$ be a monic polynomial with $f(0) = 1$. Question 1 What is the volume $c_n$ of the following set $$\{...
dummy's user avatar
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2 votes
1 answer
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Why is $\Delta(f)(s,t)= f(st)$ well-defined?

Let $G$ be a locally compact group with Haar measure $\lambda$. The locally compact group $G\times G$ then carries the Haar measure $\lambda\times \lambda$. One can then define the map $$\Delta: L^\...
Andromeda's user avatar
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3 votes
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Every Haar measure is a multiple of the counting measure.

I was asked to prove the following: Let $G$ be a group, equipped with the discrete topology. A Haar measure on $G$ is a measure $\mathcal{P}(G) \rightarrow [0, \infty]$ such that: $$\mu(K) < \...
soph6626's user avatar
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5 votes
1 answer
251 views

Are Haar measures localizable?

I'm trying to prove that Haar measures are localizable. we know that Haar measures are decomposable ( see Haar measures are decomposable) in the sense that: A measure space $(X,\mathfrak{M},\mu)$ is ...
Amirhossein Haddadian's user avatar
9 votes
0 answers
322 views

Haar measures are decomposable

In the real analysis book by Folland, section $11.1$ exercise $9$ have been come that: if $G$ is a locally compact topological group with Haar measure $\mu$, then $\mu$ is decomposable. A measure ...
Amirhossein Haddadian's user avatar
2 votes
0 answers
120 views

If the topology of locally compact topological group G is not discrete, then haar measure of singlton is zero $\mu(\{x\})=0$

in folland-real analysis,chapter 11.1, exercise $9$ have been come that: if the topology of locally compact topological group G is not discrete, then haar measure of singlton is zero meaning that $\...
Amirhossein Haddadian's user avatar
3 votes
0 answers
49 views

What is the relationship between measurable or continuos cross-sections?

Let $G$ be a locally compact Polish (or compact) group acting continuously on a locally compact Polish (or compact) space $X$, and $\mu$ a Borel measure on $X$. To be sure, continuity of the action ...
mytuh's user avatar
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1 vote
1 answer
297 views

Are Haar measures semifinite?

We know that semifinite measure space $(X,\mathcal{M},\mu)$ is a measure space that for every measurable set $E\in\mathcal{M}$ with measure $\mu(E)=\infty$, there exist a measurable set $B\subseteq E$...
Amirhossein Haddadian's user avatar

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