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^{...
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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 ...
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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 ...
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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 $H$," ...
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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 ...
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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{...
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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 ...
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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 ...
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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 $G/\Gamma$ ...
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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 $\mu(...
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$\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 ...
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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 $D=\{(u,\theta) \in (S^{n-1})^2;u\perp \theta\}$, ...
