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Let $G$ be a locally compact group (but not discrete) and let $m$ be its left Haar measure. Is it true that $\forall \epsilon$ $\exists$ $C$ such that $C$ is a compact neighborhood of the identity and the measure of $C$ is less than$\epsilon$ ?

I have little experience working with Haar measure (basically been told to just assume that it exists) so I'm wary of weird measure theoretic pathologies.

Moreover, if anyone could suggest a reference, it'd be much appreciated.

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Consider a discrete group with the counting measure... –  Mariano Suárez-Alvarez Oct 21 '11 at 0:37
    
Ah, you added your comment just as I edited to rule out the discrete case. –  Grasshopper Oct 21 '11 at 0:38
    
Hm... my inbox says there was a comment left here but I don't see it. It starts with "One of the nicest expositions of the existence and," which seems like it would have been helpful. Was it deleted? –  Grasshopper Oct 21 '11 at 2:48
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If every compact nbd of the identity $e$ has measure $\ge \epsilon$, then (by local compactness) every nbd of $e$ has measure $\ge \epsilon$. But then (by regularity) the singleton $\{e\}$ has measure $\ge \epsilon$. NOW: why does that tell you that your group is discrete? –  GEdgar Oct 21 '11 at 3:21
    
Thanks for the lead. –  Grasshopper Oct 21 '11 at 6:05

1 Answer 1

up vote 15 down vote accepted

Your technical question on Haar measure was answered in the comments, so I won't elaborate on that (unless you ask me to).


One of the nicest expositions of existence and uniqueness of Haar measure I know is the unpublished manuscript by G.K. Pedersen, The existence and uniqueness of the Haar integral on a locally compact topological group. If you're acquainted with Radon integrals, as discussed e.g. in chapter 6 of Analysis now (and read the part on invariant integration towards the end of said chapter) then you'll know most of the basics that you absolutely need to know for starting out in abstract harmonic analysis.

Alternatively, chapter 2 of Folland's A course in abstract harmonic analysis contains a thorough account of existence and uniqueness, too, and prepares the ground for further and much deeper study.

I often heard people recommend Nachbin's The Haar integral, but I admit I haven't read it myself.


Added in view of GEdgar's comment:

The basics on Radon measures are well treated in Chapter 6 of Pedersen's book mentioned above and also at the beginning of Rudin's Real and complex analysis.

Two stand-alone manuscripts on integration theory on locally compact spaces (but not specifically for groups) that I liked (I learned the basics from there):


Some classics you should be aware of:

Hewitt-Ross, Abstract Harmonic Analysis 1, gives an extremely thorough if abstract, lengthy and general treatment of topological groups and integration theory on them. The reference contains most of what you'll ever need.

Further: Loomis, Reiter and, of course, Weil. They haven't lost their value over time and can be found in every library (I'm not entirely sure that Reiter covers the existence and uniqueness theorem, though).

At some point you should definitely look at the theory of commutative groups, so I should probably also mention Rudin's Fourier Analysis on Groups.


For a quick introduction to some main facts of harmonic analysis I'd recommend George Willis's contribution to H.G. Dales et al. Introduction to Banach algebras, operators, and harmonic analysis, London Mathematical Society Student Texts, 57. Cambridge University Press, Cambridge, 2003, (e-book version), MR2060440. There are many references for further study in there.


I think that should be enough for the moment.

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Maybe I should have mentioned that Haar measure has all the good properties you can dream of... On connected groups it behaves much like Lebesgue measure on $\mathbb{R}^n$. –  t.b. Oct 21 '11 at 9:59
    
These are fine books. Suitable for someone intending to specialize in related fields. But I have the impression that the OP wanted a two-page summary or so. –  GEdgar Oct 21 '11 at 13:33
    
@GEdgar: That's what the first and last paragraphs are supposed to provide. Grasshopper asked a few questions on abstract harmonic analysis here recently, so I thought I'd give a bit more than what was asked for. Unfortunately, I don't know of a good Radon-measures-"cheat sheet", but Pedersen Ch. 6 comes close --- or do you think I should send OP to Rudin's real and complex analysis? (that's not a rhetorical question). –  t.b. Oct 21 '11 at 13:44
    
A good source of information on Radon measures (on LCH spaces) is Folland's "Real Analysis: Modern Techniques and their Applications", chapter 6 if I remember correctly. He doesn't assume that the space in question $\sigma$-compact, which some may find as an interesting generalization and others may find as a major annoyance. –  Mark Oct 21 '11 at 17:48
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Thanks a bunch for the detailed response. I've definitely got enough to work with for a while. –  Grasshopper Oct 21 '11 at 19:40

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