A non-zero positive measure on the -ring of subsets of a locally compact group generated by the family of all compact subsets, taking finite values on all compact subsets of , and satisfying either the condition of left-invariance:
where , or the condition of right-invariance:
where . Accordingly, one speaks of a left- or right-invariant Haar measure. Every Haar measure is -regular, that is,
for all .
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  (under the additional assumption that the group is separable).
If is a continuous function of compact support on , then is integrable relative to a left-invariant Haar measure on , and the corresponding integral is left-invariant (see Invariant integration), that is,
for all . A right-invariant Haar measure has the analogous property. The Haar measure of the whole group is finite if and only if is compact.
If is a left-invariant Haar measure on , then for every the following equality holds:
where is a continuous homomorphism of into the multiplicative group of positive real numbers that does not depend on the choice of the continuous function of compact support on . The homomorphism is called the modulus of ; the measure is a right-invariant Haar measure on . If , then 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 is also equivalent to the fact that every left-invariant Haar measure on is also inversely invariant, that is, for all .
If is a Lie group, then the integral with respect to a left-invariant (right-invariant) Haar measure on is defined by the formula
where the are linearly independent left-invariant (right-invariant) differential forms of order one on (see Maurer–Cartan form) and . The modulus of a Lie group is defined by the formula
where is the adjoint representation (cf. Adjoint representation of a Lie group).
Examples. 1) The Haar measure on the additive group and on the quotient group (the group of rotations of the circle) is the same as the ordinary Lebesgue measure. 2) The general linear group , or , is unimodular, and the Haar measure has the form
where for and for , and is the Lebesgue measure in the Euclidean space of all matrices of order over the field .
If is a locally compact group, is a closed subgroup of it, is the homogeneous space , and are the moduli of and , respectively, and is a continuous homomorphism of into whose restriction to is given by the formula
then there exists a positive measure on the -ring of sets that is generated by the family of compact subsets of ; it is uniquely determined by the condition:
where is any continuous function of compact support on , , and
for all continuous functions of compact support on .
|||A. Haar, "Der Massbegriff in der Theorie der kontinuierlichen Gruppen" Ann. of Math. (2) , 34 (1933) pp. 147–169|
|||N. Bourbaki, "Elements of mathematics. Integration" , Addison-Wesley (1975) pp. Chapt.6;7;8 (Translated from French)|
|||A. Weil, "l'Intégration dans les groupes topologiques et ses applications" , Hermann (1940)|
|||L.H. Loomis, "An introduction to abstract harmonic analysis" , v. Nostrand (1953)|
|||S. Helgason, "Differential geometry and symmetric spaces" , Acad. Press (1962)|
|[a1]||E. Hewitt, K.A. Ross, "Abstract harmonic analysis" , 1–2 , Springer (1979)|
Haar measure. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Haar_measure&oldid=19105