I've come across at least 3 definitions, for example:
Taken from here where $\Gamma$ is a topological group. Apparently, this definition doesn't require the Haar measure to be finite on compact sets.
Or from Wikipedia: "... In this article, the $\sigma$-algebra generated by all compact subsets of $G$ is called the Borel algebra...." Then $\mu$ defined on this sigma algebra is a Haar measure if it's outer and inner regular, finite on compact sets and translation invariant.
So I gather the important property of a Haar measure is that it's translation invariant.
Question 1: What I don't gather is, what do I get if I define it on the Borel sigma algebra as opposed to defining it on the sigma algebra generated by compact sets (as they do on Wikipedia)?
Question 2: Can I put additional assumptions on $G$ so that I can drop the requirement that $\mu$ has to be finite on compact sets?
Question 3: As you can guess from my questions I'm poking around in the dark trying to find out how to define a Haar measure suitably. Here suitably means, I want to use it to define an inner product so I can have Fourier series. Are there several ways to do this which lead to different spaces? By this I mean, if I define it on the Borel sigma algebra, can I do Fourier series for a different set of functions than when I have a measure on the sigma algebra generated by compact sets? Or what about dropping regularity? Or dropping finiteness on compact sets?
Thanks.