Invariant integration

on a group

Integration of functions on a 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 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 with respect to some 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 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), 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) is closely related to that of invariant integration.

References

[1]	N. Bourbaki, "Elements of mathematics. Integration" , Addison-Wesley (1975) pp. Chapt.6;7;8 (Translated from French)
[2]	A. Weil, "l'Intégration dans les groupes topologiques et ses applications" , Hermann (1940)
[3]	L.H. Loomis, "An introduction to abstract harmonic analysis" , v. Nostrand (1953)
[4]	E. Hewitt, K.A. Ross, "Abstract harmonic analysis" , 1 , Springer (1979)

References

[a1]	H. Reiter, "Classical harmonic analysis and locally compact groups" , Clarendon Press (1968)