Which definition of Haar measure is correct? I have encountered several different definitions of left Haar measure that don't all seem to agree.
The setting I care about is Locally Compact Groups.  
The first seems to completely disagree with the other two with the regularity dispute.
$\bf{\text{Which definition is correct? or are they all correct in different contexts?}}$
Let $G$ be a locally compact group, and by a measure I mean a Borel measure.

$\bf{\text{Definition 1:}}$
On Wikipedia: http://en.wikipedia.org/wiki/Haar_measure
A left Haar measure is a non-zero measure $\mu$ on $G$ such that
(i) $\mu(K) < \infty$ for compact $K$
(ii) $\mu$ is left $G$-invariant
(iii) $\mu$ is outer regular on Borel sets
(iv) $\mu$ is inner regular on open Borel sets
(And then an example is given to show that $\mu$ need not be inner regular on all Borel sets!)

$\bf{\text{Definition 2:}}$ 
Supplied in an answer: Why are Haar measures finite on compact sets?
A left Haar measure is a non-zero measure $\mu$ on $G$ such that
(i) $\mu$ is regular
(ii) $\mu > 0$ on open sets.
(iii) $\mu < \infty$ on compact sets.
(iv) $\mu$ is left $G$-invariant

$\bf{\text{Definition 3:}}$
Given in an appendix to a book.
A left Haar measure is a non-zero measure $\mu$ on $G$ such that
(i) $\mu$ is regular
(ii) $\mu$ is left $G$-invariant.

The first two are equivalent and both correct.  The third one requires finite-ness on compact sets.  Positivity on open sets follows from local compactness of $G$ and non-triviality of $\mu$.
