View source for Haar measure

A non-zero positive [[Measure|measure]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046060/h0460601.png" /> on the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046060/h0460602.png" />-ring <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046060/h0460603.png" /> of subsets <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046060/h0460604.png" /> of a locally compact group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046060/h0460605.png" /> generated by the family of all compact subsets, taking finite values on all compact subsets of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046060/h0460606.png" />, and satisfying either the condition of left-invariance:

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046060/h0460607.png" /></td> </tr></table>

where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046060/h0460608.png" />, or the condition of right-invariance:

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046060/h0460609.png" /></td> </tr></table>

where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046060/h04606010.png" />. Accordingly, one speaks of a left- or right-invariant Haar measure. Every Haar measure is <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046060/h04606012.png" />-regular, that is,

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046060/h04606013.png" /></td> </tr></table>

for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046060/h04606014.png" />.

A left-invariant (and also a right-invariant) Haar measure exists and is unique, up to a positive factor; this was established by A. Haar [[#References|[1]]] (under the additional assumption that the group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046060/h04606015.png" /> is separable).

If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046060/h04606016.png" /> is a continuous function of compact support on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046060/h04606017.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046060/h04606018.png" /> is integrable relative to a left-invariant Haar measure on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046060/h04606019.png" />, and the corresponding integral is left-invariant (see [[Invariant integration|Invariant integration]]), that is,

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046060/h04606020.png" /></td> </tr></table>

for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046060/h04606021.png" />. A right-invariant Haar measure has the analogous property. The Haar measure of the whole group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046060/h04606022.png" /> is finite if and only if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046060/h04606023.png" /> is compact.

If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046060/h04606024.png" /> is a left-invariant Haar measure on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046060/h04606025.png" />, then for every <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046060/h04606026.png" /> the following equality holds:

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046060/h04606027.png" /></td> </tr></table>

where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046060/h04606028.png" /> is a continuous homomorphism of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046060/h04606029.png" /> into the multiplicative group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046060/h04606030.png" /> of positive real numbers that does not depend on the choice of the continuous function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046060/h04606031.png" /> of compact support on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046060/h04606032.png" />. The homomorphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046060/h04606033.png" /> is called the modulus of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046060/h04606034.png" />; the measure <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046060/h04606035.png" /> is a right-invariant Haar measure on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046060/h04606036.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046060/h04606037.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046060/h04606038.png" /> is called unimodular; in this case a left-invariant Haar measure is also right-invariant and is called (two-sided) invariant. In particular, every compact or discrete or Abelian locally compact group, and also every connected semi-simple or nilpotent Lie group, is unimodular. Unimodularity of a group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046060/h04606039.png" /> is also equivalent to the fact that every left-invariant Haar measure <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046060/h04606040.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046060/h04606041.png" /> is also inversely invariant, that is, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046060/h04606042.png" /> for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046060/h04606043.png" />.

If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046060/h04606044.png" /> is a [[Lie group|Lie group]], then the integral with respect to a left-invariant (right-invariant) Haar measure on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046060/h04606045.png" /> is defined by the formula

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046060/h04606046.png" /></td> </tr></table>

where the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046060/h04606047.png" /> are linearly independent left-invariant (right-invariant) differential forms of order one on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046060/h04606048.png" /> (see [[Maurer–Cartan form|Maurer–Cartan form]]) and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046060/h04606049.png" />. The modulus of a Lie group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046060/h04606050.png" /> is defined by the formula

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046060/h04606051.png" /></td> </tr></table>

where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046060/h04606052.png" /> is the adjoint representation (cf. [[Adjoint representation of a Lie group|Adjoint representation of a Lie group]]).

Examples. 1) The Haar measure on the additive group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046060/h04606053.png" /> and on the quotient group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046060/h04606054.png" /> (the group of rotations of the circle) is the same as the ordinary [[Lebesgue measure|Lebesgue measure]]. 2) The [[General linear group|general linear group]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046060/h04606055.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046060/h04606056.png" /> or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046060/h04606057.png" />, is unimodular, and the Haar measure has the form

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046060/h04606058.png" /></td> </tr></table>

where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046060/h04606059.png" /> for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046060/h04606060.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046060/h04606061.png" /> for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046060/h04606062.png" />, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046060/h04606063.png" /> is the Lebesgue measure in the Euclidean space of all matrices of order <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046060/h04606064.png" /> over the field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046060/h04606065.png" />.

If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046060/h04606066.png" /> is a locally compact group, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046060/h04606067.png" /> is a closed subgroup of it, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046060/h04606068.png" /> is the [[Homogeneous space|homogeneous space]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046060/h04606069.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046060/h04606070.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046060/h04606071.png" /> are the moduli of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046060/h04606072.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046060/h04606073.png" />, respectively, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046060/h04606074.png" /> is a continuous homomorphism of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046060/h04606075.png" /> into <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046060/h04606076.png" /> whose restriction to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046060/h04606077.png" /> is given by the formula

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046060/h04606078.png" /></td> </tr></table>

then there exists a positive measure <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046060/h04606079.png" /> on the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046060/h04606080.png" />-ring <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046060/h04606081.png" /> of sets <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046060/h04606082.png" /> that is generated by the family of compact subsets of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046060/h04606083.png" />; it is uniquely determined by the condition:

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046060/h04606084.png" /></td> </tr></table>

where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046060/h04606085.png" /> is any continuous function of compact support on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046060/h04606086.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046060/h04606087.png" />, and

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046060/h04606088.png" /></td> </tr></table>

for all continuous functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046060/h04606089.png" /> of compact support on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046060/h04606090.png" />.

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  A. Haar,   "Der Massbegriff in der Theorie der kontinuierlichen Gruppen"  ''Ann. of Math. (2)'' , '''34'''  (1933)  pp. 147–169</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  N. Bourbaki,   "Elements of mathematics. Integration" , Addison-Wesley  (1975)  pp. Chapt.6;7;8  (Translated from French)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  A. Weil,   "l'Intégration dans les groupes topologiques et ses applications" , Hermann  (1940)</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  L.H. Loomis,   "An introduction to abstract harmonic analysis" , v. Nostrand  (1953)</TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top">  S. Helgason,   "Differential geometry and symmetric spaces" , Acad. Press  (1962)</TD></TR></table>



====Comments====


====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  E. Hewitt,   K.A. Ross,   "Abstract harmonic analysis" , '''1–2''' , Springer  (1979)</TD></TR></table>