Properties of Haar measure
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.
Solution 1:
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One of the nicest expositions of existence and uniqueness of Haar measure I know is the unpublished manuscript by G.K. Pedersen, The existence and uniqueness of the Haar integral on a locally compact topological group.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):
W. Arveson, Notes on Measure and Integration in Locally Compact Spaces.
O.E. Lanford III, Topics in functional analysis, Part 1: The Riesz–Markov theorem.
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.