1
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1answer
35 views

If A has positive Haar measure then $AA^{-1}$ is a neighborhood of $e$

I read the following exercise: Prove that if $G$ is a locally compact topological group with Haar measure $\mu$ and $A \subset G, \mu (A) >0$, then $AA^{-1}$ contains an open neighborhood of the ...
0
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0answers
17 views

Considering the right Haar measure on the affine group, how does the absolute value come in?

Let $ G $ be the affine group with group action defined by $ (b,a)\cdot(x,s) = (ax+b,as) $ then it is a locally compact group and as such has a Haar measure. In particular the left Haar measure is $ ...
0
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0answers
96 views

Measurable function implies equivalent to an exponential function.

This is a follow up to this question. In that question, I answered that an exponential function can be uniquely determined by three properties: a functional equation, a weak continuity assumption, and ...
0
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1answer
22 views

Haar measure of point sets

Let $G$ be a locally compact group with Haar measure $\mu$ (left or right doesn't matter to me). I know that the Haar measure is positive on open sets. What can be said about the Haar measure on ...
3
votes
0answers
44 views

Integration over uncountable set of characters

Let $G$ be a compact (assumed Hausdorff) group and $\hat{G}$ be the set of all characters of irreducible, finite-dimensional representations of $G$. It might occur that $\hat{G}$ is uncountable. It ...
4
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2answers
132 views

Intuition behind the failure of unimodularity

If $G$ is a locally compact group then up to normalization it admits a unique Haar measure: a left invariant measure defined on all Borel subsets of $G$, which assigns every compact set a finite ...
1
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0answers
27 views

Probability space associated with a compact group

Is the probability space associated with a compact group with Haar probability always a standard probability space? I recall seeing somewhere the fact that if the topology generating the Borel sigma ...
0
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1answer
64 views

The dual of L^1(G) for a locally compact group G

I might be missing something, but most literature on topological groups and harmonic analysis that I've encountered mention that $L^\infty(G)$ can be naturally identified with the dual of $L^1(G)$ by ...
2
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0answers
56 views

Abstract Fourier Analysis

I am trying to prove that given a locally compact abelian (Hausdorf) topological group, the characters on it parameterize the multiplicative linear functionals on the banach * algebra $L^1(G, d\mu)$ ...
0
votes
2answers
83 views

If $f$ is a positive function and $\int_{E}f d\lambda = 0$ then $\lambda (E) = 0$

If $f$ is a positive function and $\int_{E}f d\lambda = 0$ then $\lambda (E) = 0$ where $\lambda $ is a Haar (Radon) measure. I know that if $f$ is a positive function and $\int_{E}f d\lambda = 0$ ...
0
votes
0answers
42 views

If a normal subgroup, N, contains a lattice why does G/N have finite measure?

Suppose $G$ is a locally compact Hausdorff topological group and suppose $H \leq N \leq G$ are closed subgroups with $N$ normal. Now suppose $G/H$ has a finite $G$-invariant Boreal measure (in the ...
2
votes
1answer
79 views

Haar measure, convolution and involutions

I have some problems to follow the proof of the anti commutativity property of the convolution and involution operations defined using a Haar measure as presented in Pedersen's book "analysis Now", ...
1
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0answers
86 views

Is a regular Borel measure on a locally compact space necessarily $\sigma$-finite?

I am trying to compile a proof of the uniqueness of Haar measure. Usually this is done by multiple-integral mumbo-jumbo, abusing left and right invariance of two potential measures and invoking ...
0
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0answers
37 views

Prove uniqueness of Haar measure without using the integral?

All the literature I have seen proves the Haar measure is unique by first defining the Haar integral and then using Fubini's Theorem etc. to show that any two Haar integrals are scalar multiples of ...
3
votes
1answer
66 views

Is Steinhaus theorem ever used in topological groups?

Steinhaus theorem in $\mathbb{R}^d$ says that for $E\subset\mathbb{R}^d$ with positive measure, $E-E:=\{x-y:x,y\in E\}$ contains an open neighborhood of the origin. And for locally compact Hausdorff ...
1
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1answer
78 views

Are Compact Sets Separated In a Locally Compact Topological Group

I am studying a proof of the existence of Haar measure on locally compact groups. http://www.albanyconsort.com/HaarMeasure/HaarMeasure.pdf In this proof (at the top of page 7) when proving finite ...
1
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0answers
43 views

right multiplication by elements of a discrete subgroup preserve left haar measure?

If $\Gamma$ is a discrete subgroup of a locally compact topological group, G, it is not necessarily the case that right multiplication on $G$ by elements of $\Gamma$ will preserve a left Haar ...
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0answers
50 views

invariant measure on a quotient of a topological group

Suppose I have a locally compact topological group, $G$, and a closed subgroup $H\leq G$. Suppose $\Delta _G|_H = \Delta_H$ where the $\Delta$ are the modular functions on $G$ and $H$. How can I see ...
1
vote
1answer
168 views

A characterization of the module function on a locally compact division ring

References: Weil's Basic Number Theory(written as BNT). Bourbaki's Commutative Algebra(written as BCA). Let $K$ be a topological ring with an identity. Suppose every non-zero element of $K$ is ...
3
votes
1answer
115 views

Tricky detail in the proof of Haar's theorem

I'm trying to dig in the details of the proof of Haar's theorem, and at some point I need to use Fubini's theorem, which requires that if we want to change the order of integration over the product ...
4
votes
1answer
69 views

Which definition is correct?

I have encountered several different definitions of left Haar measure that don't all seem to agree. The setting I care about is Locally Compact Groups. The first seems to completely disagree with ...
0
votes
1answer
107 views

Haar measure $\tau$-additive?

I'm reading some results from Measure Theory Volume 4 by D.H. Fremlin, and I'm stuck on something. This is pulled out of one of his lemmas (stated more generally for topological groups): A Haar ...
4
votes
1answer
133 views

Is the Haar measure of a product of finite measure and compact, finite?

Let $G$ be a locally compact group with Haar measure $ \mu $, $K \subset G$ a compact subset and $ F \subset G $ any subset of finite Haar measure $\mu (F) < \infty $. Is the Haar measure of the ...
2
votes
1answer
138 views

On an existence of a quasi-finite left- invariant Borel measure in a non-locally compact Polish group

Let $(G,B(G))$ be a Polish group. A Borel set $A \subset G$ is called Haar null if there is a Borel probability measure $\mu$ in $G$ such that $\mu(g(A))=0$ for each $g \in G$. A Borel measure ...
7
votes
1answer
312 views

Basics of Haar measure

Suppose $G$ is a locally compact group. Then $G$ has a left-invariant measure $dg$, say, which means that $$\int f (hg) dg = \int f(g) fg$$ for any test function integrable on $G$. The ...
1
vote
1answer
199 views

Fourier transform of a measure

I'm a bit confused - How is the Fourier transform of a measure on a compact abelian group defined? specifically the Fourier transform of a measure on $\mathbb{T}$ the unit circle in the complex plain. ...
7
votes
1answer
501 views

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 $\mathcal{L}^1(G, \mu)$. ...
1
vote
0answers
329 views

Haar-measure on the torus

Good evening! Let $ \mathbb{T}:=\{ z \in \mathbb{C} ; \vert z \vert =1 \} $ be the unit circle in the complex plane. We denote the trace Borel-$\sigma$-algebra on $\mathbb{T}$ by ...
5
votes
1answer
134 views

Measure of a conjugacy class in a compact group

Suppose $G$ is a compact group endowed with Haar measure $\mu$. If $g \in G$, then denote by $g^G$ the conjugacy class of $g$ in $G$. Is there anything that can be said in general about $\mu(g^G)$? ...
3
votes
0answers
187 views

If $G$ is a locally compact Hausdorff group, when does $G/Z$ have a probability Haar measure?

I am reading an introductory material about topological groups and the question in the tittle comes up. Due this Proposition Proposition. A locally compact Hausdorff topological group $G$ is ...
3
votes
1answer
129 views

Restricted Direct Products in Koch's Number Theory

On p.353 of Number Theory: Algebraic Numbers and Functions by Helmut Koch, he considers a group $G$ which is the restricted direct product of the locally compact abelian groups $G_i$ with respect to ...