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Questions tagged [locally-compact-groups]

Locally compact groups are among the main objects of study of abstract harmonic analysis. They arise e.g. in number theory, abstract Fourier analysis, representation theory, Lie theory, operator theory, etc. Their main property is the existence and uniqueness of Haar measure. Please combine with relevant other tags whenever appropriate in order to reflect the main intentions of the question in the tags.

Locally compact groups are among the main objects of study of abstract harmonic analysis. They arise e.g. in number theory, abstract Fourier analysis, representation theory, Lie theory, operator theory, etc. Their main property is the existence and uniqueness of Haar measure. Please combine with relevant other tags whenever appropriate in order to reflect the main intentions of the question in the tags.

463 questions
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Properties of Haar measure

Let $G$ be a locally compact group (but not discrete) and let $m$ be its left Haar measure. Is it true that $\forall \epsilon$ $\exists$ $C$ such that $C$ is a compact neighborhood of the identity and the measure of $C$ is less than$\epsilon$ ? I…
Grasshopper
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21
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A net version of dominated convergence?

Let $G$ be a locally compact Hausdorff Abelian topological group. Let $\mu$ be a Haar measure on $G$, i.e. a regular translation invariant measure. Let $f$ be fixed in $\mathcal{L}^1(G, \mu)$. Define the function from $G$ to $\mathcal{L}^1(G)$…
Jeff
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Existence of orthonormal basis for $L^2(G)$ in $C_c(G)$.

Suppose that $G$ is a locally compact (Hausdorff) group endowed with the Haar measure. It is well-known that the compactly supported functions $C_c(G)$ are dense in $L^2(G)$. In the book "Operator algebras, theory of $C^{*}$-algebras and von…
Calculix
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Theorem of Steinhaus

The Steinhaus theorem says that if a set $A \subset \mathbb R^n$ is of positive inner Lebesgue measure then $\operatorname{int}{(A+A)} \neq \emptyset$. Is it true that also $\operatorname{int}{(tA+(1-t)A)} \neq \emptyset$ for $t \in(0,1)$? It is…
Richard
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Translation-invariant metric on locally compact group

Let $G$ be a locally compact group on which there exists a Haar measure, etc.. Now I am supposed to take such a metrisable $G$, and given the existence of some metric on $G$, prove that there exists a translation-invariant metric, i.e., a metric $d$…
Lit
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Distributions over locally compact Abelian groups: when can they be Fourier transformed?

Pontryagin duality shows us every locally compact Abelian group---such as $\mathbb{R}^n$, $\mathbb{Z}$, the circle $\mathbb{R}/\mathbb{Z}$ or any finite Abelian group---has a Fourier transform. In the case of $\mathbb{R}$, it follows from …
Juan Bermejo Vega
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Local-Global Principle and the Cassels statement.

In a recent article I have read, i.e. " Lecture notes on elliptic curves ", Prof.Cassels remarks in page-110 that There is not merely a local-global principle for curves of genus-$0$, but it has a quantitative formulation ( and also, …
IDOK
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12
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2 answers

Factorization of maps between locally compact Hausdorff space

Let me consider a space to be locally compact, if every point has a neighborhood whose closure is compact. Consider a continuous map $f:X\to Y$ between locally compact Hausdorff spaces. Is it true that $f$ can be factored as…
gurgur dilon
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subgroup of connected locally compact group

I need a reference or a short proof for the following property: A nontrivial connected locally compact group $G$ contains an infinite abelian subgroup.
Zouba
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Stone-Weierstrass implies Fourier expansion

To prove the existence of Fourier expansion, I have to solve the following exercise, which supposedly follows from the Stone-Weierstrass theorem: Let $G$ be a compact abelian topological group with Haar measure $m$. Let $\hat G$ be the dual. The…
Chera
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Haar measure on O(n) or U(n)

Every locally compact group has left-invariants haar measures. In particular, the compact groups $\operatorname{O}(n)$ and $\operatorname{U}(n)$ have them. I was wondering if there is a realization of such a measure on these groups, or its integral…
Henrique Tyrrell
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measure of open set with measure Haar

By a Haar measure on a locall compact group (Hausdorff) we mean a positive measure $\mu$ (contains the borel set's) such that The measure $\mu$ is left invariant The measure μ is finite on every compact set Is $\mu$-regular (i.e. outer and inner…
user126033
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Young's inequality for discrete convolution

Young's inequality for convolution of functions states that for $f\in L^p(\mathbb{R}^d)$ and $g\in L^q(\mathbb{R}^d)$ we have $$\|f\star g\|_r\le\|f\|_p\|g\|_q$$ for $p$, $q$, $r$ satisfying $$\frac{1}{p}+\frac{1}{q}=\frac{1}{r}+1.$$ Does this…
mpiktas
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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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Haar measure of quotient group

Suppose $G$ is a (Hausdorff) compact group with normalised Haar measure $\mu$, and that $H\trianglelefteq G$ is a closed normal subgroup. Is it true that the pushforward of $\mu$ to $G/H$ is the normalised Haar measure of $G/H$? That is, is it true…
Cronus
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