All Questions
Tagged with topological-groups haar-measure
46 questions
6
votes
0
answers
105
views
Dependence on Urysohn's Lemma in Cartan's Construction of Haar Measure
This question was posted by someone else on stackexchange three months ago, but no one has answered as of yet:
Cartan's 1940 paper, Sur la mesure de Haar, claims to provide a proof of the existence ...
- 85
0
votes
1
answer
64
views
Transitive map on a profinite group
Let $f$ be a continuous endomorphism of a compact Hausdorff totally disconnected topological group $G$ and let $H$ be a closed normal subgroup of G such that $f(H)\subseteq H$ and with $\mu(H)=0$ ...
- 361
1
vote
0
answers
154
views
measure of Haar
Let $(G,K)$ be a Gelfand pair.
Why, for a function $f$ $K$-binvariant with respect to a compact subgroup $K$ of a group $G$, do we have the following equality:
$$ f(xy) = \int_K f(xky) \, dk$$
A ...
- 109
5
votes
1
answer
229
views
Does any locally compact topological group which is not Hausdorff have a Haar measure?
In 'Linear Analysis and Representation Theory' by Steven Gaal at the end of Chapter IV, page 227, the author claims that any locally compact topological group $G$ which is not Hausdorff has a Haar ...
- 360
0
votes
0
answers
38
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Are closures of products of unimodular subgroups unimodular?
Let $G$ be a locally compact group, $N \subset G$ a unimodular normal subgroup, and $H \subset G$ a discrete (hence unimodular) subgroup. Does it follow that the closure $\overline{NH} \subset G$ is ...
- 4,164
1
vote
0
answers
277
views
Using the von Neumann crossed product to introduce a measure on the orbit space?
Suppose we're given an action (possibly: ergodic) of a group G (say, $\mathbb{R}$) on a measure space $(X, \mu)$ (possibly: a standard probability space).
Question: is there a natural way of using the ...
- 101
7
votes
2
answers
320
views
Uniqueness of left-invariant Borel probability measure on compact groups
On a compact topological group, consider two left-invariant probability measures $\mu$ and $\nu$ defined on the Borel sigma-algebra. Is it true that they coincide?
It is classical that the Haar ...
2
votes
1
answer
257
views
Haar measures of compact subgroups
Let $G$ be a locally compact group, $K$ a compact subgroup in $G$, and $\mu_K$ the normalized Haar measure on $K$:
$$
\mu_K(K)=1.
$$
Let us denote by $\widetilde{\mu_K}$ the measure on $G$ defined as ...
- 7,426
20
votes
0
answers
333
views
Existence of orthonormal basis for $L^2(G)$ in $C_c(G)$
Remark: I cross-posted this question on MSE and added a bounty to it.
Suppose that $G$ is a locally compact (Hausdorff) group endowed with the Haar measure. It is well-known that the compactly ...
- 321
15
votes
1
answer
497
views
For what LCH groups is the Haar measure $\mu(U x U)$ bounded?
Let $G$ be a locally compact Hausdorff (LCH) topological group with left Haar measure $\mu$. Given a compact unit neighborhood $U$, consider the function
$$
\Phi: \quad G \to (0,\infty), \quad x \...
- 734
1
vote
1
answer
647
views
Haar measure coming from Pontryagin duality v/s Fourier inversion
Not research but advertising this question from mse in case someone wants to answer.
I'm struggling with some bookkeeping associated with the Pontryagin duality theorem. I'm thinking about the first ...
- 675
5
votes
0
answers
194
views
Haar mesure on $\mathrm{GL}_{d}(F)$
$\DeclareMathOperator\GL{GL}$Let $F$ be a $\mathfrak{p}$-adic field and $\mathscr{O}_{F}$ its valuation ring. For any measurable subset of $M_{d}(F)$ such as
$$
A=
\left( \begin{array}{ccc}
a_{11}+t^{\...
- 479
8
votes
1
answer
318
views
A group where the Weil topology induced by the Haar measure does not coincide with the original topology
Let $(G,\tau)$ be a locally compact Hausdorff topological group that is $\sigma$-finite with respect to the Haar measure $\mu:\mathcal{B}(G)\to[0,\infty]$ ($\mathcal{B}(G)$ is the Borel $\sigma$-...
3
votes
0
answers
110
views
Is this a lattice?
Let $R$ be a locally compact ring (commutative with unit) and let $D\subset R$ be a discrete cocompact subring (cocompact means the additive group $R/D$ is compact). Let $G$ be a semisimple linear ...
user130903
0
votes
0
answers
97
views
How large this subset is to say that it should equal the group?
Let $\alpha$ be a continuous automorphism on a compact group $G$ with normalized Haar measure $m$. We may say $\alpha$ is $n$-splitting, if the set
$$\text{Spl}_n(\alpha):=\left\{g\in G: \prod_{k=1}^...
- 2,118
3
votes
0
answers
178
views
Question about regular representation of compact group
I first define the setting for my question. Let $G$ be a compact group with probability Haar measure $\mu_G$. Denote by $\lambda$ the left regular representation on $L^2(G)$ defined for $f \in L^2(G)$ ...
- 429
5
votes
0
answers
202
views
Invariant measure on coset space and integrable functions
Let $G$ be a locally compact abelian group, and $H$ a closed subgroup. Let $C_c(G)$ be the space of continuous, compactly supported complex valued functions on $G$. General theory of Haar measure ...
- 6,180
1
vote
0
answers
206
views
Extrinsic applications of Haar measure
I am looking for examples of (not necessarily deep) results whose proofs rely on the Haar measure (or rather such that there exists an "elegant" proof involving it). Further, the formulation of such ...
- 111
1
vote
0
answers
295
views
Invariant measures on locally compact homogeneous spaces
Edit: answered in the first comment. This turned out to be really easy, as the orbits of an open $\sigma$-compact subgroup yield a partition of $X$ into open $\sigma$-compact subsets.
Let $G$ be a ...
- 373
13
votes
1
answer
576
views
Are Hausdorff measures on the real line Haar measures for some locally compact topology?
For $0\leq d\leq 1$, let $\lambda_d$ be the $d$-dimensional Hausdorff measure on $\mathbb{R}$. Note that it is translation-invariant. Does there exist a locally compact topology $\mathscr{T}_d$ on $\...
- 32.5k
4
votes
1
answer
384
views
Invariant integration on principal bundles
Let $G$ be a sufficiently nice topological or Lie group (e.g. compact), and let $H$ be a closed subgroup. This data determines a principal $H$ bundle $G \rightarrow G/H$ defined by the projection $g \...
- 143
1
vote
1
answer
236
views
Definition of Haar integral in Bushnell and Henniart
In Bushnell and Henniart's The Local Langland's Conjecture for GL(2) they define a right Haar integral on a locally profinite group $G$ as being a non-zero linear functional
$$
I: C^{\infty}_{c}(G) \...
- 183
0
votes
1
answer
311
views
Measure on group invariant under group action on metric space
This is a question very similar to I recently asked on mathexchange, but different enough to get its own entry in MO.
The setting is still the same. I consider the metric space $\mathbb{R}$ and the ...
- 103
3
votes
1
answer
77
views
Convergence of some object depending on functions with compact support
Let $G$ be a locally compact group with unimodular Haar measure $\mu$. We consider the Hilbert space $\mathscr{H}:= L_{\mu}^2(G)$ together with the unitary representation $\pi : G \to U(\mathscr{H})$ ...
- 429
2
votes
0
answers
102
views
Is this concrete set generically Haar-null?
This question is related to this problem on MO about generically Haar-null sets in locally compact Polish groups but is more concrete.
First we recall the definition of a generically Haar-null set in ...
- 41.8k
5
votes
0
answers
214
views
On generically Haar-null sets in the real line
First some definitions.
For a Polish space $X$ by $P(X)$ we denote the space of all $\sigma$-additive Borel probability measures on $X$. The space $P(X)$ carries a Polish topology generated by the ...
- 41.8k
3
votes
1
answer
141
views
Measure on orbits of $N$ under conjugation by $H$
Let $G$ be a locally compact topological group with closed subgroups $H, N$ and $H$ normalizing $N$. Then $H$ acts continuously on $N$ by conjugation. If it will help, assume that $N$ is nilpotent, ...
- 6,180
1
vote
1
answer
297
views
Haar measure, can image of modular function be any subgroup of $(0,\infty)$?
It is easy to find examples of locally compact second countable Hausdorff topological groups $G$ whose modular function $\Delta$ has image $\{1\}$ or $(0,\infty)$. Are there groups $G$ of this kind ...
- 282
3
votes
1
answer
347
views
Subgroups of finite non-zero Haar measure of abelian locally compact groups
Is it true that every subgroup of finite non-zero Haar measure of an abelian locally compact group should be open and compact? This is obviously true for the case of discrete abelian groups. Thanks.
- 895
3
votes
1
answer
197
views
Is the sumset of two Haar positive closed subsets of a Polish group non-meager?
A famous Steinhaus theorem says that if measurable subsets $A,B$ of a locally compact topological group $G$ have positive Haar measure, then the difference $AA^{-1}$ is a neighborhood of the unit and ...
- 41.8k
5
votes
1
answer
588
views
Why is it possible to normalize the Haar measure on the quotient?
I just asked a question which is related to the one I'm about to ask, but I realized my question can be reduced to the following: let $G$ be a locally compact abelian group with Haar measure $\mu$, ...
- 6,180
1
vote
1
answer
228
views
Structure of locally compact non discrete topological division algebras without the use of Haar measure
There is a well-known structure theorem for locally compact non discrete topological division algebras, see here
https://math.stackexchange.com/q/1160086/187521
(I repost it here because I think it ...
- 255
5
votes
1
answer
297
views
If $ F(x,\bullet) \in {L^{\infty}}(G,B) $ for all $ x \in G $, then is $ x \mapsto F(x,\bullet) $ strongly measurable?
Let $ (X,\Sigma,\mu) $ be a $ \sigma $-finite measure space and $ B $ a Banach space. A function $ f: X \to B $ is said to be strongly $ \mu $-measurable iff it is the almost-everywhere pointwise ...
- 1,396
3
votes
0
answers
739
views
Justification of the convolution operation of $L^1(G)$ functions where $G$ is a LCA group (measurability)
Suppose $G$ is a locally compact abelian Hausdorff group (LCA), and $\lambda$ is the Haar measure on it. We all know the convolution of two $L^1(\lambda)$ functions $f$ and $g$ on $G$ is defined as
$$...
- 960
11
votes
1
answer
667
views
Compact Quantum Groups and the Existence of the Classical Haar Measure
Before I state my question, let me provide the definition of a compact quantum group.
Definition: An ordered pair $ \mathscr{G} = (\mathscr{A},\Phi) $ is called a compact quantum group if
$ ...
user36116
10
votes
1
answer
467
views
Haar measurable sets and quotient maps
Let $G$ be a locally compact Hausdorff group with a Haar measure $\mu$, let $H$ be a closed normal subgroup of $G$, and let $q: G \to G/H$ be the quotient homomorphism. Let $\nu$ be a Haar measure ...
- 101
12
votes
3
answers
1k
views
To what extent has the Haar measure been generalized?
It is known that all locally compact groups, and therefore compact groups, have a left-invariant Haar measure which is unique up to scalar constant, also a right-invariant one. Is there a strictly ...
- 561
1
vote
2
answers
635
views
Mean value theorems for the Haar integral?
Let $G$ be a compact topological group (feel free to add hypotheses if necessary). Is there any mean value theorem for its (normalized to 1) Haar integral?
In general, are there mean value theorems ...
- 5,407
1
vote
1
answer
278
views
Harmonic Analysis [closed]
Let $G$ be a locally compact group, $H$ be a closed subgroup and $N$ be a normal subgroup of $G$ such that $H\subseteq N$. How can we get $$\int_{G/H} f(xH)d...
- 11
3
votes
0
answers
279
views
Almost conjugation-invariant neighborhoods of units in locally compact groups
Let $G$ be a locally compact topological group with unity $e$ and left Haar measure $m$. Let also $g\in G$ be a given element and $U$ a neighborhood (of compact closure) of $e$.
I am interested to ...
7
votes
2
answers
2k
views
Haar measure for large locally compact groups
In this answer, Gerald Edgar mentions that Haar measure is naturally defined on the $\sigma$-algebra of Baire sets (the smallest $\sigma$-algebra that contains all the compact $G_\delta$ sets) of a ...
- 44.4k
1
vote
0
answers
220
views
exotic compact group
Let $G$ be compact (and Hausdorff) group, $\mu$ be Haar measure on $G$. Is it always true that $(G,\mu)$ is a standard probability space (Lebesgue-Rokhlin space)? It is so if (a priori not iff) the ...
- 109k
6
votes
2
answers
3k
views
How to define the quotient of a measure which is invariant under group action?
I am a physicist, and I have the following problem. Consider a locally compact group G acting over a measure space $(X, {\cal B}, \mu)$, and assume that $\mu$ is G-invariant. My problem is how to "...
- 339
8
votes
1
answer
1k
views
Must a locally compact group be Hausdorff in order to possess a Haar measure?
Does the existence of (left) Haar measure on a locally compact topological group require that the group be Hausdorff?
- 2,814
8
votes
4
answers
9k
views
Haar Measure on a Quotient [closed]
Suppose you have a locally compact group G with a discrete subgroup H. Of course G has a unique (up to scalar) Haar measure, but it seems that G/H has and induced Haar measure as well.
How does ...
- 89
19
votes
9
answers
6k
views
Haar measure on a quotient, References for
I remember reading Weil's "Basic Number Theory" and giving up after a while. Now I find myself thinking of it (thanks to some comments by Ben Linowitz).
Right from the very beginning, Weil ...
- 7,442