View source for Invariant integration

<!--
i0522401.png
$#A+1 = 75 n = 0
$#C+1 = 75 : ~/encyclopedia/old_files/data/I052/I.0502240 Invariant integration
Automatically converted into TeX, above some diagnostics.
Please remove this comment and the {{TEX|auto}} line below,
if TeX found to be correct.
-->

{{TEX|auto}}
{{TEX|done}}

''on a group''

Integration of functions on a [[Topological group|topological group]] that has a certain invariant property with respect to the group operations. Thus, let  $  G $
be a locally compact topological group, let  $  C _ {0} ( G) $
be the vector space of all continuous complex-valued functions with compact support on  $  G $
and let  $  I $
be an integral on  $  C _ {0} ( G) $, 
that is, a positive [[Linear functional|linear functional]] on  $  C _ {0} ( G) $(
$  I f \geq  0 $
for  $  f \geq  0 $). 
The integral  $  I $
is called left-invariant (or right-invariant) if  $  I ( gf  ) = If $(
or  $  I ( fg ) = If  $) 
for all  $  g \in G $, 
$  f \in C _ {0} ( G) $; 
here

$$ 
( gf  ) ( x )  =  f ( g  ^ {-} 1 x ) ,\ \ 
( fg ) ( x)  =  f ( x g ) .
$$

The integral  $  I $
is called (two-sided) invariant if it is both left- and right-invariant. The mapping  $  I \rightarrow \widehat{I}  $, 
where  $  \widehat{I}  f = I \widehat{f}  $, 
$  \widehat{f}  ( x) = f ( x  ^ {-} 1 ) $, 
defines a one-to-one correspondence between the classes of left- and right-invariant integrals on  $  C _ {0} ( G) $. 
If  $  I = \widehat{I}  $, 
then  $  I $
is called inversion invariant.

There exists on every locally compact group  $  G $
a non-zero left-invariant integral; it is unique up to a numerical factor (the Haar–von Neumann–Weil theorem). This integral is called the left Haar integral. The following equation holds:

$$ 
I ( fg )  =  \Delta ( g ) I f ,
$$

where  $  g \in G $, 
$  f \in C _ {0} ( G ) $
and  $  \Delta $
is a continuous homomorphism from the group  $  G $
into the multiplicative group of positive real numbers (a positive character). Furthermore,  $  \widehat{I}  f = I ( f / \Delta ) $. 
The character  $  \Delta $
is called the modulus of  $  G $. 
If  $  \Delta ( g) \equiv 1 $, 
then  $  G $
is called unimodular. In this case  $  I $
is a two-sided invariant integral.

In particular, every compact group (where  $  I1 < \infty $, 
$  \widehat{I}  = I $) 
and every discrete group (where  $  If = \sum _ {g} f ( g) $, 
$  f \in C _ {0} ( G) $) 
is unimodular.

According to the Riesz theorem, every integral on  $  C _ {0} ( G) $
is a [[Lebesgue integral|Lebesgue integral]] with respect to some [[Borel measure|Borel measure]]  $  \mu $
which is uniquely defined in the class of Borel measures that are finite on each compact subset  $  K \subset  G $. 
The left- (or right-) invariant measure  $  \mu $
corresponding to the left (right) Haar integral on  $  C _ {0} ( G) $
is called the left (right) [[Haar measure|Haar measure]] on  $  G $.

Let  $  H $
be a closed subgroup of  $  G $
and let  $  \Delta _ {0} $
be the modulus of  $  H $. 
If  $  \Delta _ {0} $
can be extended to a continuous positive character of  $  G $(
cf. [[Character of a group|Character of a group]]), then there exists on the left homogeneous space  $  X = G / H $
a relatively invariant integral  $  J $, 
that is, a positive functional on the space  $  C _ {0} ( X) $
of continuous functions with compact support on  $  X $
that satisfies the identity

$$ 
J ( gf  )  =  \delta ( g ) J f
$$

for all  $  g \in G $, 
$  f \in C _ {0} ( X) $; 
here

$$ 
( gf  ) ( x)  =  f ( g  ^ {-} 1 x ) ,\ \ 
\delta ( g)  =  
\frac{\Delta _ {0} ( g) }{\Delta ( g) }
 ,
$$

and  $  \Delta $
is the modulus of  $  G $. 
This integral is defined by the rule  $  J f = I ( \delta \widetilde{f}   ) $, 
where  $  I $
is the left Haar integral on  $  G $
and  $  \widetilde{f}  $
is a function on  $  G $
such that

$$ 
f ( gH )  =  I _ {0} (( g \widetilde{f}   ) _ {H} ) .
$$

( $  I _ {0} $
is the left Haar integral on  $  H $
and  $  \phi _ {H} $
is the restriction of  $  \phi $
to  $  H $.) 
This is well-defined since  $  \widetilde{f}  \rightarrow f $
is a mapping from  $  C _ {0} ( G) $
onto  $  C _ {0} ( X) $
and  $  Jf = 0 $
when  $  f = 0 $. 
The notion of an invariant mean (cf. [[Invariant average|Invariant average]]) is closely related to that of invariant integration.

====References====
<table><TR><TD valign="top">[1]</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">[2]</TD> <TD valign="top">  A. Weil,   "l'Intégration dans les groupes topologiques et ses applications" , Hermann  (1940)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  L.H. Loomis,   "An introduction to abstract harmonic analysis" , v. Nostrand  (1953)</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  E. Hewitt,   K.A. Ross,   "Abstract harmonic analysis" , '''1''' , Springer  (1979)</TD></TR></table>

====Comments====

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  H. Reiter,   "Classical harmonic analysis and locally compact groups" , Clarendon Press  (1968)</TD></TR></table>