Tagged Questions
4
votes
1answer
48 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
36 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
55 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
75 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 ...
6
votes
1answer
145 views
Basics of Haar measure
I feel totally confused about the sentence "therefore, ... because $dgh^{-1}$ is another left-invariant measure." What is the reason for "therefore"? Why is $dgh$ a left-invariant measure? (It seems ...
1
vote
1answer
88 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.
...
6
votes
1answer
289 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
216 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 ...
3
votes
1answer
96 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
166 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 ...
2
votes
1answer
116 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 ...
