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.

  • 1
    $\begingroup$ What book? The definition should be whatever you need to prove that Haar measures are unique up to scale. If you have enough properties to prove that then they're necessarily enough to imply all other properties. $\endgroup$ May 15 '13 at 6:04
  • 1
    $\begingroup$ Of course you are correct. I realize now that I had the incorrect definition of regular. With the correct definition of regular, all definitions above are the same. $\endgroup$
    – roo
    May 15 '13 at 20:08
  • 1
    $\begingroup$ @Kyle Since you answered your question. Don't hesitate to answer it. We want to have the less unanswered questions as possible :) $\endgroup$
    – Norbert
    May 22 '13 at 1:48

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 $\mu$.

  • 4
    $\begingroup$ For some authors, "finite on compact sets" might be part of the definition of the word "regular". $\endgroup$ Jun 9 '16 at 14:43
  • $\begingroup$ It is also very often part of the definition of "Borel measure". $\endgroup$
    – Cronus
    Aug 19 '16 at 0:27

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