Invariant integration

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on a group

Integration of functions on a topological group that has a certain invariant property with respect to the group operations. Thus, let be a locally compact topological group, let be the vector space of all continuous complex-valued functions with compact support on and let be an integral on , that is, a positive linear functional on ( for ). The integral is called left-invariant (or right-invariant) if (or ) for all , ; here

The integral is called (two-sided) invariant if it is both left- and right-invariant. The mapping , where , , defines a one-to-one correspondence between the classes of left- and right-invariant integrals on . If , then is called inversion invariant.

There exists on every locally compact group 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:

where , and is a continuous homomorphism from the group into the multiplicative group of positive real numbers (a positive character). Furthermore, . The character is called the modulus of . If , then is called unimodular. In this case is a two-sided invariant integral.

In particular, every compact group (where , ) and every discrete group (where , ) is unimodular.

According to the Riesz theorem, every integral on is a Lebesgue integral with respect to some Borel measure which is uniquely defined in the class of Borel measures that are finite on each compact subset . The left- (or right-) invariant measure corresponding to the left (right) Haar integral on is called the left (right) Haar measure on .

Let be a closed subgroup of and let be the modulus of . If can be extended to a continuous positive character of (cf. Character of a group), then there exists on the left homogeneous space a relatively invariant integral , that is, a positive functional on the space of continuous functions with compact support on that satisfies the identity

for all , ; here

and is the modulus of . This integral is defined by the rule , where is the left Haar integral on and is a function on such that

( is the left Haar integral on and is the restriction of to .) This is well-defined since is a mapping from onto and when . The notion of an invariant mean (cf. Invariant average) is closely related to that of invariant integration.


[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)



[a1] H. Reiter, "Classical harmonic analysis and locally compact groups" , Clarendon Press (1968)
How to Cite This Entry:
Invariant integration. D.P. Zhelobenko (originator), Encyclopedia of Mathematics. URL:
This text originally appeared in Encyclopedia of Mathematics - ISBN 1402006098