# Tagged Questions

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 ...

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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 ...
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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 $\mathcal{L}^1(G, \mu)$. ...
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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.
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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 ...
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### Functorial construction of the convolution algebra of measures on a group

Let $G$ be a lcoally compact group and $C_c(G) = \lim_{K \subset G} C(K)$ its space of continuous functions with compact support endowed with the topology of the limit of banach spaces $C(K)$ with $K$ ...
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### Haar measure on a profinite group is the inverse limit of the counting measure on its quotients?

I've heard this a few times now, though I've never seen a precise result. I guess the precise statement would be close to: Let $N_i$ be a basis normal subgroup neighborhoods of the identity in a ...
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### Showing a measure on a locally compact group is left invariant

I am trying to verify that the measure $\frac{1}{|x|}dx$ is a Haar measure on $\mathbb{R}\backslash \{0\}$. For every open interval $(a_{n},b_{n})\subset \mathbb{R}$ not containing $0$, I have that ...
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### Uniqueness of Haar Measures

Haar measure on a LCA (locally compact abelian) group $G$ is said to be unique ... up to a scaling factor. This is not as elegant as it might be expected because this requires a choice of a unit ...
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### Can Haar Measures Exist On Not Locally Compact Spaces?

In a reading course on measure theory this semester I had the pleasure of preparing a lecture covering the existence-uniqueness of Haar measure on locally compact groups. Since the proofs (as ...
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### Is there a formula for the Haar measure on a product of groups?

Let $(G_{n})_{n \in \mathbb{N}}$ be a sequence of locally compact topological groups with a corresponding sequence $(\mu_{n})_{n \in \mathbb{N}}$ of Haar measures. Is there a way to construct a ...
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### Bounded approximate identities in one-sided ideals of $L_1(G)$

Let $G$ be a locally compact group and consider its $L_1$-group algebra, that is, $L_1(G)$. It is easily seen that $L_1(G)$ has a two-sided bounded approximate identity: just use a basis of compact ...
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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 ...
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### Why is $L^{1}(G)$ unital if and only if $G$ is discrete?

I've seen it stated in several sources and lecture notes for Abstract Harmonic Analysis that for a locally compact group $G$, $L^{1}(G)$ is unital if and only if $G$ is discrete. What about the ...
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### Locally compact infinite dihedral group

Is there any topology other than discrete which can be given to an infinite dihedral group to make it locally compact topological group? If no, why is it so?
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### Non-discrete locally compact group

Why every non-discrete locally compact group contains a nontrivial convergent sequence?
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### If both $H$ and $G/H$ are locally compact then $G$ is locally compact (topological Group)

How do I prove this statement? Let $G$ be a Topological group and let $H$ be a subgroup of $G$, if both $H$ and $G/H$ are locally compact then $G$ is locally compact. (we will endow the set $G/H$ ...
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### Why are characters required to be continuous?

I learned from several places that in defining a character of a topological group $G$, we often require it to be continuous, i.e. $\omega:G\to \mathbb{C}^{\times}$ is a continuous group homomorphism. ...
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### Pontryagin dual of the unit circle

Define the unit circle as $\frac{\mathbb{R}}{2\pi\mathbb{Z}}.$ I know the Pontryagin dual (looking at properties of the Fourier transform on locally compact Abelian groups) is $\mathbb{Z}$ but why? ...
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### question on subgroup of compact group

Suppose $G$ is a compact metrizable abelian group, is it true that $G$ has no finite index subgroups iff $G$ is connected? Any reference or help is appreciated. Thanks in advance! Here are my ...
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### Unimodular groups

Let F be a non-archimedean local fields of characteristic p (for p any given prime number): the field of formal Laurent series Fq((T)) over a finite field Fq (where q is a power of p). Is GL(n,F) ...
Let $k$ be the completion of an algebraic number field at a prime divisor $\mathfrak{p}$. We note that $k$ is locally compact. Let $k^{+}$ be the additive group of $k$ which is a locally compact ...