# 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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### A subgroup of full measure is dense given a haar measure

I want to know why if $\mu$ is a haar measure on a compact $G$ and $\mu(A)=\mu(G)$ then $A$ is dense in $G$. This fact is mentioned in the wikipedia page, but I couldn't find a proof for it.
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### Convolution on a locally compact group is associative

Consider the following fragment from Folland's book "A course in abstract harmonic analysis" (question is below the image). Can someone explain why the boxed equality is true? Don't we need ...
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### Inherence of full measure sets from a Lie group to its homogeneous space.

Let $G$ be a Lie group with Haar measure $\mu$ and $\Gamma$ be a lattice of $G$. Namely $\Gamma$ is a closed discrete subgroup of $G$ with $G/\Gamma$ admitting a left $G$-invariant probability measure ...
1 vote
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### Estimating the distance from 1 to the conmutator of two elements in a Lie group.

I am reading Thurston's book (Three dimensional geometry and topology), and there is something I can't understand. Let $G$ be a Lie Group, in the book it says that there exists an $\epsilon > 0$ ...
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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 ...
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### Question on integration on a local field

Let $F$ be a non-Archimedean local field, and $\mu_F$ a Haar measure on $F$. The space $C^{\infty}_c(F)$ of locally constant functions of compact support is spanned by characteristic functions of the ...
1 vote
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### How to write a Haar measure on $SO(n)$ and $SU(n)$ given Haar measure on $GL(n)$?

How to write a Haar measure on $SO(n)$ and $SU(n)$ given Haar measure on $GL(n)\,$? I know that $GL(n)$ has the Haar measure $(\det(A)^{-1} dA)$. I try to write from these a Haar measure for the ...
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