# 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 Measures of $U(d)$ and $O(d)$

I'm trying to compute the Haar measures of finite unitary group $U(d)$ and finite orthogonal group $O(d)$. I've shown that both are compact groups. Most of computings I've seen base on Peter-Weyl ...
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### Reference for $p$-adic Haar integral

I have stumbled upon the notion of a $K$-valued Haar integral on a locally compact group, where $K$ is a non-Archimedean field, as well as the $K$-valued modular function, in an article of Schikhof. ...
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### Connection between Haars measure Hilbertspace and Quantum mechanics.

Iam a physics student,I don't have any background in measuring theory. What I know is group theory and some priliminary basics of topology. In a book by MV Altaisky I see I don't understand how he ...
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### Riemannian volume vs Haar measure?

Consider $SL(d,\mathbb{R})$ or $GL(d,\mathbb{R})$. Is it true that the Riemannian volume measure $\mathrm{vol}$ on these Lie groups as submanifolds of $\mathbb{R}^{d \times d}$ is the same as the Haar ...
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### Existence of set of positive Haar measure containing torus

I am struggling around with the following problem, and was hoping someone could help. Let $G$ be a non-abelian compact connected Lie group with Haar measure $\nu$. Fix $h\in G$ and consider the ...
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### $\mathrm{d}\mu(x) = \mathrm{d}\lambda(x)/x$, $\lambda$ is Lebesgue. Is $L^{\infty}([0,1],\lambda) = L^{\infty}([0,1],\mu)$?

Definitions: Let $\lambda$ be the Lebesgue measure on $G$. Define Haar measure $\mu$ on $G$ by $\mathrm{d}\mu(x) := \frac{1}{x}\mathrm{d}\lambda(x)$. $G$ is multiplicative group, $G=(R_+,⋅)$. Since ...
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### Haar measure from Cartan metric?

Consider the following $d$-dimensional Lie algebra, $$[X_i, X_j] = f_{ij}^k X_k$$ so that the Cartan metric is given by $$g_{ij} = f_{ia}^bf_{jb}^a$$ Now, what I wish to know is whether the Cartan ...
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### Volume element of Haar measure on SO(3) with Euler angle parametrization

I have a real-valued function $f(\alpha, \beta, \gamma)$ which takes Euler angles $\alpha, \beta, \gamma$ as input, that I would like to average over the uniform distribution on orientations of 3-...
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### Haar measure on SU(n)

I can't seem to find a formula for Haar integral $$\int_{SU(n)} F(U) d \mu$$ The integration will probably be executed using some parametrization with $n^2 -1$ parameters as I believe $SU(n)$ is ...
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### Compute haar measure of a subset of SU(n)

Let $$X_n=\{(a,b,c)\in (SU(n))^{3} : c=aba^{-1}b^{-1}\}\subset SU(n)$$ with $SU(n)$ munished with its standard metric (say, normalized so that the total volume is $1$). Is there a good method to ...
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### The history of Haar measure

Haar measure is a well-known concept in measure theory. Many books are perfectly dedicated to present its existence and uniqueness such as measure theory for D. Cohn. I am looking for a good ...
Let $\mu$ be a Borel probability measure on $\operatorname{SL}_3(\mathbb{R}) / \operatorname{SL}_3(\mathbb{Z})$. Also let $g \in \operatorname{SL}_3 (\mathbb{R})$ and $h_\mu (g)$ be the measure ...