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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.

The Haar measure assigns an "invariant volume" to subsets of locally compact topological groups, consequently defining an integral for functions on those groups.

Haar measures are used in analysis, number theory, group theory, representation theory, statistics, probability theory, and ergodic theory.

371 questions
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For which topological measure spaces do open sets always have positive measure?

I want to generalize the result for $\mathbb{R}^n$, that any non-trivial open set has positive measure. Open sets and measure zero Question: For which types of topological measure spaces is it true that every non-trivial (i.e. non-empty) open set…
Chill2Macht
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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 space $(X,\mathfrak{M},\mu)$ is decomposable if: (i)…
Amirhossein Haddadian
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Uniform distribution on the unit sphere rotated by a random orthogonal matrix

Question 1. Let $u\in \mathbb{R}^n $ be a random vector uniformly distributed on $\mathbb{S}^{n-1}$, and $T\in \mathbb{R}^{n\times n}$ be a random orthogonal matrix. If $u$ and $T$ are independent, is $Tu$ uniformly distributed on $S^{n-1}$ and…
Nuno
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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 exists a $G$-invariant Radon measure $\mu$ on…
Carlos Esparza
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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 constructed a measure which is finite and Radon. However…
rfloc
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How to compute the volume of a ball in the space of matrices with norm distance as a metric? What about unitary or special unitary matrices?

Consider the space of $n \times n$ complex matrices with metric equal to the distance in the Frobenius norm so that $d(A,B) \equiv \| A - B \| \equiv \sqrt{Tr[(A-B)^{\dagger}(A-B)]}$. I want to know the volume of a ball defined as $B_{\epsilon}(U_0)…
Sherif F.
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Is the Haar measure on Abelian groups regular?

I have just started studying Fourier analysis on topological groups from Folland's A course in abstract harmonic analysis and Rudin's Fourier analysis on groups. It seems that Rudin defines the Haar measure as a regular measure on the Borel sets,…
sami.spricht.sprache
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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 $S$. Is $T$ itself necessarily a translation…
xir
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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 on $SO(3)$, and the most common one seem to be…
Severin
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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 : Is $M$ measurable? On the positive side, I have…
Moinsdeuxcat
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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. $G=\biguplus_{\gamma\in\varGamma}\varOmega\gamma$. Claim: If $…
user7610
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Dependence on Urysohn's lemma in Cartan's construction of Haar measure

Cartan's 1940 paper, Sur la mesure de Haar, claims to provide a proof of the existence and uniqueness of Haar measure for locally compact abelian groups without the use of axiom of choice. However, Cartan claimed the existence of a partition of…
Chen
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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 $\chi^{j}$ are the characters of the irreducible unitary spin-j…
user674359
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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; apologies for the bad name—I do not mean,…
MCS
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A question about André Weil’s converse to Haar’s Theorem on the existence of Haar measures

Let $ (G,\cdot,e) $ be a group, and suppose that there are a $ \sigma $-ring $ \Sigma $ on $ G $ and a measure $ \mu: \Sigma \to [0,\infty] $, non-trivial, such that the following properties hold: $ \Sigma $ is left-invariant w.r.t. $ \cdot $,…
Transcendental
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