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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:
+
A non-zero positive [[Measure|measure]] $ \mu $ on the $ \sigma $-ring $ M $ of subsets $ E $ of a locally compact group $ G $ generated by the family of all compact subsets, taking finite values on all compact subsets of $ G $, and satisfying either the condition of '''left-invariance''':
 +
$$
 +
\forall E \in M, ~ \forall g \in G: \qquad
 +
\mu(E) = \mu(g E),
 +
$$
 +
where $ g E = \{ g x \in G \mid x \in E \} $, or the condition of '''right-invariance''':
 +
$$
 +
\forall E \in M, ~ \forall g \in G: \qquad
 +
\mu(E) = \mu(E g),
 +
$$
 +
where $ E g = \{ x g \in G \mid x \in E \} $. Accordingly, one speaks of a left- or right-invariant Haar measure. Every Haar measure is $ \mu $-'''regular''', i.e.,
 +
$$
 +
\forall E \in M: \qquad
 +
\mu(E) = \sup(\{ \mu(K) \in \mathbf{R}_{\geq 0} \mid K \subseteq E ~ \text{and} ~ K ~ \text{is a compactum} \}).
 +
$$
  
<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>
+
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 $ G $ is separable).
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046060/h0460608.png" />, or the condition of right-invariance:
+
If $ f \in {C_{c}}(G) $, then $ f $ is integrable relative to a left-invariant Haar measure on $ G $, and the corresponding integral is [[Invariant integration|left-invariant]], i.e.,
 +
$$
 +
\forall g_{0} \in G: \qquad
 +
\int_{G} f(g) ~ \mathrm{d}{\mu(g)} = \int_{G} f(g_{0} g) ~ \mathrm{d}{\mu(g)}.
 +
$$
 +
A right-invariant Haar measure has the analogous property. The Haar measure of the whole group $ G $ is finite if and only if $ G $ is compact.
  
<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>
+
If $ \mu $ is a left-invariant Haar measure on $ G $, then the following equality holds:
 +
$$
 +
\forall f \in {C_{c}}(G), ~ \forall g_{0} \in G: \qquad
 +
\int_{G} f(g g_{0}^{-1}) ~ \mathrm{d}{\mu(g)} = \Delta(g_{0}) \int_{G} f(g) ~ \mathrm{d}{\mu(g)},
 +
$$
 +
where $ \Delta $ is a continuous homomorphism of $ G $ into the multiplicative group $ \mathbf{R}^{+} $ of positive real numbers that does not depend on the choice of $ f $. The homomorphism $ \Delta $ is called the '''modular function''' of $ G $; the measure $ \Delta(g^{-1}) ~ \mathrm{d}{\mu(g)} $ is a right-invariant Haar measure on $ G $. If $ \Delta(g) = 1 $ for all $ g \in G $, then $ G $ 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 $ G $ is also equivalent to the fact that every left-invariant Haar measure $ \mu $ on $ G $ is also inversely invariant, i.e., $ \mu(E^{-1}) = \mu(E) $ for all $ E \in M $.
  
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,
+
If $ G $ is a [[Lie group|Lie group]], then the integral with respect to a left-invariant (right-invariant) Haar measure on $ G $ is defined by the formula
 +
$$
 +
\int_{G} f(x) ~ \mathrm{d}{\mu(x)} = \int_{G} f ~ \omega_{1} \wedge \cdots \wedge \omega_{n},
 +
$$
 +
where the $ \omega_{i} $’s are linearly independent left-invariant (right-invariant) differential forms of order $ 1 $ on $ G $ (see the [[Maurer–Cartan form|Maurer–Cartan form]]) and $ n = \dim(G) $. The modular function of a Lie group $ G $ is defined by the formula
 +
$$
 +
\forall x \in G: \qquad
 +
\Delta(x) = |\! \det(\operatorname{Ad} x)|,
 +
$$
 +
where $ \operatorname{Ad} $ is the [[Adjoint representation of a Lie group|adjoint representation]].
  
<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>
+
'''Examples.'''
 +
# The Haar measure on the additive group $ \mathbf{R} $ and on the quotient group $ \mathbf{R} / \mathbf{Z} $ (the group of rotations of the circle) is the same as the ordinary [[Lebesgue measure|Lebesgue measure]].
 +
# The [[General linear group|general linear group]] $ \operatorname{GL}(n,\mathbf{F}) $, where $ \mathbf{F} \in \{ \mathbf{R},\mathbf{C} \} $, is unimodular, and the Haar measure has the form
 +
$$
 +
\mathrm{d}{\mu(x)} = |\! \det(x)|^{- k} ~ \mathrm{d}{x},
 +
$$
 +
where $ k = n $ for $ \mathbf{F} = \mathbf{R} $ and $ k = 2 n $ for $ \mathbf{F} = \mathbf{C} $, and $ \mathrm{d}{x} $ is the Lebesgue measure on the Euclidean space of all matrices of order $ n $ over the field $ \mathbf{F} $.
  
for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046060/h04606014.png" />.
+
If $ G $ is a locally compact group, $ H $ is a closed subgroup of it, $ X $ is the [[Homogeneous space|homogeneous space]] $ G / H $, $ \Delta $ and $ \delta $ are the modular functions of $ G $ and $ H $, respectively, and $ \chi $ is a continuous homomorphism of $ G $ into $ \mathbf{R}^{+} $ whose restriction to $ H $ is given by the formula
 +
$$
 +
\forall h \in H: \qquad
 +
\chi(h) = \delta(h) \Delta(h^{-1}),
 +
$$
 +
then there exists a positive measure $ \nu $ on the $ \sigma $-ring $ T $ of sets $ E \subseteq G / H = X $ that is generated by the family of compact subsets of $ X $; it is uniquely determined by the condition:
 +
$$
 +
\forall f \in {C_{c}}(G): \qquad
 +
\int_{G / H} \left[ \int_{H} f(g h) ~ \mathrm{d}{\mu(h)} \right] \mathrm{d}{\nu(g)} = \int_{G} f(g) \chi(g) ~ \mathrm{d}{\mu(g)},
 +
$$
 +
where $ g = g H \in X $, and
 +
$$
 +
\forall h \in {C_{c}}(X): \qquad
 +
\int_{X} h(g^{-1} x) ~ \mathrm{d}{\nu(x)} = \chi(g) \int_{X} h(x) \mathrm{d}{\nu(x)}.
 +
$$
  
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).
+
====References====
 
 
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>
+
<table>
 
+
<TR><TD valign="top">[1]</TD><TD valign="top">
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:
+
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">
<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>
+
N. Bourbaki, “Elements of mathematics. Integration”, Addison-Wesley (1975), Chapt. 6–8. (Translated from French)</TD></TR>
 
+
<TR><TD valign="top">[3]</TD> <TD valign="top">
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
+
A. Weil, “L’intégration dans les groupes topologiques et ses applications”, Hermann (1940).</TD></TR>
 
+
<TR><TD valign="top">[4]</TD> <TD valign="top">
<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>
+
L.H. Loomis, “An introduction to abstract harmonic analysis”, v. Nostrand (1953).</TD></TR>
 
+
<TR><TD valign="top">[5]</TD> <TD valign="top">
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" />.
+
S. Helgason, “Differential geometry and symmetric spaces”, Acad. Press (1962).</TD></TR>
 +
</table>
  
 
====References====
 
====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====
 
  
 
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<table>
====References====
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<TR><TD valign="top">[a1]</TD> <TD valign="top">
<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>
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E. Hewitt, K.A. Ross, “Abstract harmonic analysis”, '''1–2''', Springer (1979).</TD></TR>
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</table>

Latest revision as of 20:18, 23 April 2017

A non-zero positive measure $ \mu $ on the $ \sigma $-ring $ M $ of subsets $ E $ of a locally compact group $ G $ generated by the family of all compact subsets, taking finite values on all compact subsets of $ G $, and satisfying either the condition of left-invariance: $$ \forall E \in M, ~ \forall g \in G: \qquad \mu(E) = \mu(g E), $$ where $ g E = \{ g x \in G \mid x \in E \} $, or the condition of right-invariance: $$ \forall E \in M, ~ \forall g \in G: \qquad \mu(E) = \mu(E g), $$ where $ E g = \{ x g \in G \mid x \in E \} $. Accordingly, one speaks of a left- or right-invariant Haar measure. Every Haar measure is $ \mu $-regular, i.e., $$ \forall E \in M: \qquad \mu(E) = \sup(\{ \mu(K) \in \mathbf{R}_{\geq 0} \mid K \subseteq E ~ \text{and} ~ K ~ \text{is a compactum} \}). $$

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 ([1]) (under the additional assumption that the group $ G $ is separable).

If $ f \in {C_{c}}(G) $, then $ f $ is integrable relative to a left-invariant Haar measure on $ G $, and the corresponding integral is left-invariant, i.e., $$ \forall g_{0} \in G: \qquad \int_{G} f(g) ~ \mathrm{d}{\mu(g)} = \int_{G} f(g_{0} g) ~ \mathrm{d}{\mu(g)}. $$ A right-invariant Haar measure has the analogous property. The Haar measure of the whole group $ G $ is finite if and only if $ G $ is compact.

If $ \mu $ is a left-invariant Haar measure on $ G $, then the following equality holds: $$ \forall f \in {C_{c}}(G), ~ \forall g_{0} \in G: \qquad \int_{G} f(g g_{0}^{-1}) ~ \mathrm{d}{\mu(g)} = \Delta(g_{0}) \int_{G} f(g) ~ \mathrm{d}{\mu(g)}, $$ where $ \Delta $ is a continuous homomorphism of $ G $ into the multiplicative group $ \mathbf{R}^{+} $ of positive real numbers that does not depend on the choice of $ f $. The homomorphism $ \Delta $ is called the modular function of $ G $; the measure $ \Delta(g^{-1}) ~ \mathrm{d}{\mu(g)} $ is a right-invariant Haar measure on $ G $. If $ \Delta(g) = 1 $ for all $ g \in G $, then $ G $ 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 $ G $ is also equivalent to the fact that every left-invariant Haar measure $ \mu $ on $ G $ is also inversely invariant, i.e., $ \mu(E^{-1}) = \mu(E) $ for all $ E \in M $.

If $ G $ is a Lie group, then the integral with respect to a left-invariant (right-invariant) Haar measure on $ G $ is defined by the formula $$ \int_{G} f(x) ~ \mathrm{d}{\mu(x)} = \int_{G} f ~ \omega_{1} \wedge \cdots \wedge \omega_{n}, $$ where the $ \omega_{i} $’s are linearly independent left-invariant (right-invariant) differential forms of order $ 1 $ on $ G $ (see the Maurer–Cartan form) and $ n = \dim(G) $. The modular function of a Lie group $ G $ is defined by the formula $$ \forall x \in G: \qquad \Delta(x) = |\! \det(\operatorname{Ad} x)|, $$ where $ \operatorname{Ad} $ is the adjoint representation.

Examples.

  1. The Haar measure on the additive group $ \mathbf{R} $ and on the quotient group $ \mathbf{R} / \mathbf{Z} $ (the group of rotations of the circle) is the same as the ordinary Lebesgue measure.
  2. The general linear group $ \operatorname{GL}(n,\mathbf{F}) $, where $ \mathbf{F} \in \{ \mathbf{R},\mathbf{C} \} $, is unimodular, and the Haar measure has the form

$$ \mathrm{d}{\mu(x)} = |\! \det(x)|^{- k} ~ \mathrm{d}{x}, $$ where $ k = n $ for $ \mathbf{F} = \mathbf{R} $ and $ k = 2 n $ for $ \mathbf{F} = \mathbf{C} $, and $ \mathrm{d}{x} $ is the Lebesgue measure on the Euclidean space of all matrices of order $ n $ over the field $ \mathbf{F} $.

If $ G $ is a locally compact group, $ H $ is a closed subgroup of it, $ X $ is the homogeneous space $ G / H $, $ \Delta $ and $ \delta $ are the modular functions of $ G $ and $ H $, respectively, and $ \chi $ is a continuous homomorphism of $ G $ into $ \mathbf{R}^{+} $ whose restriction to $ H $ is given by the formula $$ \forall h \in H: \qquad \chi(h) = \delta(h) \Delta(h^{-1}), $$ then there exists a positive measure $ \nu $ on the $ \sigma $-ring $ T $ of sets $ E \subseteq G / H = X $ that is generated by the family of compact subsets of $ X $; it is uniquely determined by the condition: $$ \forall f \in {C_{c}}(G): \qquad \int_{G / H} \left[ \int_{H} f(g h) ~ \mathrm{d}{\mu(h)} \right] \mathrm{d}{\nu(g)} = \int_{G} f(g) \chi(g) ~ \mathrm{d}{\mu(g)}, $$ where $ g = g H \in X $, and $$ \forall h \in {C_{c}}(X): \qquad \int_{X} h(g^{-1} x) ~ \mathrm{d}{\nu(x)} = \chi(g) \int_{X} h(x) \mathrm{d}{\nu(x)}. $$

References

[1] A. Haar, “Der Massbegriff in der Theorie der kontinuierlichen Gruppen”, Ann. of Math. (2), 34 (1933), pp. 147–169.
[2] N. Bourbaki, “Elements of mathematics. Integration”, Addison-Wesley (1975), Chapt. 6–8. (Translated from French)
[3] A. Weil, “L’intégration dans les groupes topologiques et ses applications”, Hermann (1940).
[4] L.H. Loomis, “An introduction to abstract harmonic analysis”, v. Nostrand (1953).
[5] S. Helgason, “Differential geometry and symmetric spaces”, Acad. Press (1962).

References

[a1] E. Hewitt, K.A. Ross, “Abstract harmonic analysis”, 1–2, Springer (1979).
How to Cite This Entry:
Haar measure. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Haar_measure&oldid=41190
This article was adapted from an original article by D.P. ZhelobenkoA.I. Shtern (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article