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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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
66 views

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
79 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
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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
1 vote
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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69 views

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
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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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35 views

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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45 views

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 $...
dmh's user avatar
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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
143 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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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, ...
Ronald J. Zallman's user avatar
3 votes
1 answer
66 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 $...
Ronald J. Zallman's user avatar
1 vote
0 answers
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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
109 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
26 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 ...
Master.AKA's user avatar
1 vote
0 answers
88 views

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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Uniqueness of action invariant measure on orbits

Let $G$ be a locally compact group with normalized Haar probability measure $\mu$ and $X$ a locally compact Hausdorff space, $\phi: G \times X \rightarrow X$ a continuous action and $x \in X$. We ...
Carson James's user avatar
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Something similar to the Minkowski's lattice theorem for the locally compact groups (problem with an exercise)

There is the following problem in the book "Fourier Analysis on Number Fields" by D. Ramakrishnan and R. Valenza: Let $G$ be a locally compact abelian subgroup with Haar measure $\mu$, and $\...
filipux's user avatar
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1 vote
0 answers
35 views

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
70 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
  • 460
3 votes
0 answers
144 views

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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0 answers
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Problem concerning Haar measures on locally compact Hausdorff groups

Before I state what my problem is I first want to give some context. A Haar measure is a measure on the Borel subsets of a locally compact Hausdorff group $X$. The Haar measure is inner regular on ...
somethingsomething69's user avatar
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0 answers
71 views

Help with the Haar measure on locally compact hausdorff groups

Before I state what my problem is I first wanna give some context. A haar measure is a measure on the borel subsets of a locally compact hausdorff group X. The Haar measure is inner regular on open ...
somethingsomething69's user avatar
5 votes
1 answer
225 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
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0 answers
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Does $ch(K)\ast \phi =0$ implies $K=0$ (convolution product on locally compact group)?

Let $G$ be a locally compact group and $K$ an open subgroup. Let $\mu$ be a Haär measure. Let $ch(K)$ be the characteristic function on $K$ and $\phi\in L^{2}(G)$ such that $||\phi||_{2}\neq 0$. Does $...
Marsault Chabat's user avatar
9 votes
0 answers
301 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
114 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
42 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
244 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
5 votes
0 answers
69 views

Lifting the quotient homogeneous space to the group and define the measure

Let $G$ be a Lie group equipped with a Haar measure $\mu$ and $H$ be a closed subgroup of $G$, equipped with a Haar measure $\nu$. We do not assume that there exist a $G$-invariant measure on $H\...
taylor's user avatar
  • 557
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0 answers
60 views

Expectation w.r.t. Haar measure of $R\cdot\text{diag}(R^\top \Sigma R) \cdot R^\top$

Let $\Sigma \in \mathbb{R}^{n \times n}$ be a positive definite matrix and $\mu$ be the Haar measure over all orthogonal matrices $\in \mathbb{R}^{n\times n}$. What is $$\mathbb{E}_{R \sim \mu} \left[...
tobayes's user avatar
  • 21
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0 answers
21 views

the mutual information between the column vectors of a Haar unitary

Given a Haar random unitary $U=(U_1, \cdots, U_d) \in \mathbb{C}^{d\times d}$ where $U_i$ is the $i$-th column vector of $U$, what about the mutual information between $U_i$ and $U_j$ for any $i \ne j$...
cyrie wang's user avatar
1 vote
1 answer
80 views

Intuition for $\int_G\chi_V=\dim(V^G)$?

Let $G$ be a compact topological group with Haar measure $dg$. We can decompose a finite-dimensional complex representation as $V\cong \mathbb C^k\oplus V_1\oplus\dots\oplus V_n$, where $G$ acts ...
Nikhil Sahoo's user avatar
  • 1,590
3 votes
0 answers
72 views

Simple expression for Haar measure of $U(N)$ and $SU(n)$

The Haar measure for the group of real $n \times n$ invertible matrices has a simple expression: $$\mathrm d\mu=\dfrac{\mathrm dA}{|\det(A)|^n}\qquad\color{blue}{(1)}$$ The invariance under group ...
proton's user avatar
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1 vote
0 answers
172 views

Haar measures are saturated

We know that a saturated measure is a measure space $(X,\mathcal{M},\mu)$ such that $\mu:\mathcal{M}\rightarrow [0,\infty]\;$ and $\;\mathcal{M}=\widetilde{\mathcal{M}}\;$ where $\;\widetilde{\...
Amirhossein Haddadian's user avatar
0 votes
0 answers
47 views

Generating random but non-uniform quantum state

This is a question motivated from quantum computing but is also relevant to math. I would like an algorithm that generates a random quantum state (that is, an element of the complex projective space ...
nervxxx's user avatar
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2 votes
0 answers
56 views

Is the bi-invariant Haar measure also invariant under any automorphism of $\text{SL}(n,\mathbb R)$?

Let $G=\text{SL}(n,\mathbb R)$, a unimodular Lie group and let $\mu$ be a (the) bi-invariant Haar measure on it. Let $\alpha$ be any automorphism on this group. I wonder if $\alpha_*\mu=\mu$ By the ...
taylor's user avatar
  • 557
8 votes
2 answers
149 views

Measure Preserving Self-Map of Compact Abelian Group Commuting with Ergodic Translation

Let $K$ be a compact abelian group with its probability Haar measure, and let $S:K\to K$ be an ergodic translation automorphism. Suppose that $T:K\to K$ is a measure-preserving map that commutes with $...
xir's user avatar
  • 240
4 votes
2 answers
231 views

If the pushforward of $\mu$ under the group action is invariant, then $\mu$ must be the Haar measure.

Let $G=\text{PU}(d)$ the projective unitary group, acting on the complex projective space $X=\mathbb{P}(\mathbb{C}^d)$, in the usual way. Let $\mu:\mathcal{B}(G)\to[0,\infty]$ be a Radon measure, ...
Saúl Pilatowsky-Cameo's user avatar
1 vote
0 answers
36 views

"Haar measures" on CGWH groups?

Is there a suitable notion of "Haar measure" on CGWH groups (topological groups whose topology is compactly generated weak Hausdorff)? I know that if a topological group admits a Haar ...
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