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.

123 questions
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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 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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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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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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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 ...
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:  ...