Editing Haar measure (section)

==Haar's theorem==
There is, [[up to]] a positive multiplicative constant, a unique [[countably additive]], nontrivial measure <math>\mu</math> on the Borel subsets of <math>G</math> satisfying the following properties:

* The measure <math>\mu</math> is left-translation-invariant: <math>\mu(gS) = \mu(S)</math> for every <math>g\in G</math> and all Borel sets <math>S\subseteq G</math>.
* The measure <math>\mu</math> is finite on every compact set: <math>\mu(K) < \infty</math> for all compact <math>K \subseteq G</math>.
* The measure <math>\mu</math> is  [[outer regular]] on Borel sets <math>S\subseteq G</math>: <math display="block"> \mu(S) = \inf \{\mu(U): S \subseteq U, U \text{ open}\}.</math>
* The measure <math>\mu</math> is [[inner regular]] on open sets <math>U\subseteq G</math>: <math display="block"> \mu(U) = \sup \{\mu(K): K \subseteq U, K \text{ compact}\}.</math>

Such a measure on <math>G</math> is called a ''left Haar measure.''  It can be shown as a consequence of the above properties that <math>\mu(U)>0</math> for every non-empty open subset <math>U\subseteq G</math>. In particular, if <math>G</math> is compact then <math>\mu(G)</math> is finite and positive, so we can uniquely specify a left Haar measure on <math>G</math> by adding the normalization condition <math>\mu(G)=1</math>.

In complete analogy, one can also prove the existence and uniqueness of a ''right Haar measure'' on <math>G</math>. The two measures need not coincide.

Some authors define a Haar measure on [[Baire set]]s rather than Borel sets. This makes the regularity conditions unnecessary as Baire measures are automatically regular. [[Paul Halmos|Halmos]]<ref name=":0">{{cite book| last1=Halmos |first1=Paul R.| title=Measure theory |date=1950|publisher=Springer Science+Business Media |location=New York |isbn=978-1-4684-9442-6| page=219-220}}</ref> uses the nonstandard term "Borel set" for elements of the [[sigma-ring|<math>\sigma</math>-ring]] generated by compact sets, and defines Haar measures on these sets.

The left Haar measure satisfies the inner regularity condition for all [[sigma-finite|<math>\sigma</math>-finite]] Borel sets, but may not be inner regular for ''all'' Borel sets. For example, the product of the [[unit circle]] (with its usual topology) and the [[real line]] with the [[discrete topology]] is a locally compact group with the [[product topology]] and a Haar measure on this group is not inner regular for the closed subset <math>\{1\} \times [0,1]</math>. (Compact subsets of this vertical segment are finite sets and points have measure <math>0</math>, so the measure of any compact subset of this vertical segment is <math>0</math>. But, using outer regularity, one can show the segment has infinite measure.)

The existence and uniqueness (up to scaling) of a left Haar measure was first proven in full generality by [[André Weil]].<ref>{{Citation | last = Weil | first = André | author-link = André Weil | title = L'intégration dans les groupes topologiques et ses applications | series = Actualités Scientifiques et Industrielles | publisher = Hermann | year = 1940 | place = Paris | volume = 869}}</ref> Weil's proof used the [[axiom of choice]] and [[Henri Cartan]] furnished a proof that avoided its use.<ref>{{Citation | last = Cartan | first = Henri | author-link = Henri Cartan | title = Sur la mesure de Haar | journal = [[Comptes rendus de l'Académie des sciences|Comptes Rendus de l'Académie des Sciences de Paris]] | volume = 211 | pages = 759–762 | year = 1940}}</ref> Cartan's proof also establishes the existence and the uniqueness simultaneously. A simplified and complete account of Cartan's argument was given by [[Erik Alfsen|Alfsen]] in 1963.<ref>{{Citation | last = Alfsen | first = E.M. | title = A simplified constructive proof of existence and uniqueness of Haar measure | journal = Math. Scand. | volume = 12 | pages = 106–116 | year = 1963 | doi = 10.7146/math.scand.a-10675 | url=http://www.mscand.dk/article/view/10675/8696}}</ref> The special case of invariant measure for [[Second-countable space|second-countable]] locally compact groups had been shown by Haar in 1933.<ref name="Haar"/>