# Tagged Questions

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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 $L^1(G, d\mu)$ ...
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### 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 show that $\lambda (E) = 0$ where $\lambda$ is a Haar (Radon) measure. I know that if $f$ is a positive function and ...
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### 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 ...
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### 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", ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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. ...
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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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### 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 ...
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### 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 $\mu(g^G)$? ...
### 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 ...
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 ...