Talk:Haar measure

message: It might be interesting to add this to the article. 134.184.26.82 (talk) 12:34, 7 October 2020 (UTC)
thread_title: Relationship to Jeffrey's prior

message: I think some of the facts on Pontryagin duality really belong here.
Charles Matthews 08:20, 17 May 2004 (UTC)
thread_title: Duality

message: TeX isn't working.  Argh! 
CSTAR 15:24, 6 Jun 2004 (UTC)
thread_title: TeX

message: I forgot to say in my change to the Haar measure article that I removed the statement of the theorem because it shows up twice. Oleg Alexandrov 04:54, 17 Jan 2005 (UTC)
thread_title: The theorem

message: The article defines Haar measure on the sigma algebra generated by compact sets. However, some classic books like Rudin's "Fourier Analysis on Groups" consider Haar measure over the sigma algebra generated by all open sets. Rudin's book does not prove explicitly the existance of such a measure. Is there a mistake on Rudin's book or Haar measure can be defined over the sigma algebra generated by all open sets? I think this is a very important matter since it has consequences on the mesurability of the continuos functions. Can someone provide a clarification about this matter and maybe an example that shows that one needs to restrict the construction to the sigma algebra generated by compact sets or a reference to a book where the measure is constructed over the sigma algebra generated by all open sets? That would be a great contribution to this article! —Preceding unsigned comment added by 201.231.28.30 (talk • contribs) 
: No, Rudin's book is correct, but so is the WP article.  The statement given in the article is exactly as given in Halmos' book "Measure Theory" (now admittedly klunky, but authoritative and completely accessible).  Haar measure can also be approached via Radon measures on locally compact spaces; these are non-negative functionals on the space of continuous functions of compact support.  There is a unique (up to a scalar) non-zero left invariant Radon measure on a locally compact group.  Now Radon measures can be "extended" in a canonical way to the σ-algebra of Borel sets (generated by open sets). See Rudin's "Real and Complex Analysis 2.14" (Big Rudin 1966) for a precise statement and the proof. The delicate point here is uniqueness, which requires somewhat different regularity properties for the measures in the statements.
:Most of these technical distinctions are immaterial for metrizable separable locally compact groups or compact groups. This would be massive technical overkill for the article to elaborate further, I think.
:--CSTAR 20:32, 11 September 2006 (UTC)
thread_title:  More on Borel sets 

message: The theorem says that \mu(B) > 0 for any nonempty Borel set B.  What if B is a singleton set?  This is ridiculous.  128.208.1.224 (talk) 21:26, 4 March 2009 (UTC)
replies: text: The theorem says that \mu(B) > 0 for any open nonempty Borel set B.  If a singleton set is open, then indeed it must have positive measure, and the group is discrete. JackSchmidt (talk) 22:50, 4 March 2009 (UTC)
thread_title:  The theorem is stated incorrectly. 

message: Could someone explicitly discuss the Harr measure on unitary matrices?  It is very important in physics (especially for SU(N) in QFT and unitary transformations in quantum information theory).  I haven't found anywhere it is explained well. Njerseyguy (talk) 16:14, 15 July 2009 (UTC)
thread_title:  Harr measure on unitary matrices 

message: In section "Existence and uniqueness of the left Haar measure" it is said that
<<It turns out that there is, up to a positive multiplicative constant, only one left-translation-invariant countably additive regular measure μ on the Borel subsets of G such that μ(U) > 0 for any open non-empty Borel set U. Such a measure is called a left Haar measure>>
While in section "Uses" it is stated that
<<A frequently used technique for proving the existence of a Haar measure on a locally compact group G is showing the existence of a left invariant Radon measure on G.>>
Why bother to "use a technique for proving the existence of a Haar measure" if it always exists? Where am I missing the point?  —Preceding unsigned comment added by 88.1.123.97 (talk) 17:03, 6 November 2009 (UTC)
thread_title:  Existence of the Haar measure 

message: There is a split and an expand tag. The section as it stands cannot be split because a split article would not be viable. The expand tag seems reasonable so I have removed the split tag. Op47 (talk) 15:09, 6 April 2012 (UTC)
thread_title:  Split 

message: Can someone say what expansion they would like to see in this section?
The statements under Haar integral seem quite clear.
If the question is 'How to define the Haar integral', or another such
question, then this is properly a question about the definition of the 'Lebesgue integral' with respect to a given measure. Also there are
links to 'Lebesgue integration' AND 'indicator functions' here.
Further, with the 4 examples of Haar measures beneath this heading, it seems that there is enough meat in this section for someone who longs for some computational examples of Haar integrals.
99.33.255.116 (talk) 22:42, 2 June 2013 (UTC)
thread_title:  Suggested need for expansion of section 'Haar integral' 

message: The measure for open submanifolds is stated to be dnx/J(x). But shoulden't it be d^nx / |\det(J(x))| instead? --Jobu0101 (talk) 09:22, 14 December 2016 (UTC)
thread_title:  Open submanifolds 

message: Quote from the section about existence:
: The function {\displaystyle \mu _{A}} is additive on disjoint compact subsets of {\displaystyle G}, which implies that it is a regular content.
I understand that this is a content. I don't see, however, how this is a regular content. Additivity for disjoint compact sets is part of the definition of content. But in order to get a Radon measure from it, we need a regular content. I couldn't find anywhere a convincing proof that the so-constructed content is regular. Could someone who understands this please expand upon that claim? --138.246.2.201 (talk) 12:13, 28 July 2017 (UTC)
thread_title:  Existence of Haar measure: *regular* content? 

message: The section titled Examples contains this paragaph:
"* The unit hyperbola \{(x,y) : x^2 - y^2 = 1 , x > 0 \} can be taken as a group under multiplication defined as with split-complex numbers z = x + y j, \ \ j^2 = +1 . The usual area measure in the crescent C = \{(x,y) : |y| < x, \ \ x^2 - y^2 < 1 \} serves to define hyperbolic angle as the area of its hyperbolic sector. The Haar measure of the unit hyperbola is generated by the hyperbolic angle of segments on the hyperbola. For instance, a measure of one unit is given by the segment running from (1,1) to (e,1/e), where e is Euler's number. Hyperbolic angle has been exploited in mathematical physics with rapidity standing in for classical velocity."
Unfortunately the points (1,1) and (e,1/e) do not lie on the stated hyperbola, but instead on the hyperbola given by 
{(x,y) in R2  |  xy  =  1}.
I hope that someone familiar with this example can fix this.
In addition, this article has many places where it could be phrased a lot more clearly.50.205.142.50 (talk) 22:00, 8 June 2020 (UTC)
thread_title:  An unstated change of hyperbola in the middle of a paragraph