If $X$ is a locally compact Hausdorff space (l.c.H. space) then the Riesz representation theorem says that positive linear functionals on $C_c(X)$ are given by Borel measures on $X$. Unfortunately there is not a one-to-one correspondence, and so one introduces some class of "regular" measures on $X$ and show that for these measures there is a one-to-one correspondence. These measures are then usually called Radon. However, there is not a unique way to do this, and so Radon measure means different things for different authors. Now to my knowledge the two most used definitions of Radon are:
- Finite on compact sets, inner regular on open sets, and outer regular on Borel sets.
- Finite on compact sets, and inner regular on Borel sets.
Finally, for a l.c.H. group $G$ one defines Haar measure as an invariant Radon measure, and one can show that this is unique up to a constant, and if we want left or right invariance. Now, my question is the following. For Haar measure I have only seen definition 1. of Radon being used. So is it possible to also use 2.? And does this make things harder/easier or just the same?