68 L. CLOZEL, SPECTRAL THEORY OF AUTOMORPHIC FORMS18.1, 18 .6]. Thus it has a natural Borel structure; we will consider measures on G
relative to this Borel structure^3.
If G is compact any unitary representation (n, H) of G - by convention all our
Hilbert spaces are separable - is a Hilbertsum :H =EB n Hn
where n runs over an (at most) countable set and Hn is an irreducible representa-
tion. If G is not compact there is an analogous decomposition as a direct integral.
First of all, there is a notion of a measurable family ( f--> He, ( ( E G) of
Hilbert spaces. Here He, is a Hilbert space depending on(; it is supposed to vary
measurably in (; this is defined by the requirement that there are sufficiently many
vector fields ( f--> v( () E He, such that the functions llv( ()lie, are measurable ... see
[22, A 96 and A 69]. So this is essentially a "measurable bundle of Hilbert spaces".
Clearly the definition will apply to any set Z endowed with a Borel structure.
If each He, carries a representation of G, and ( f--> He, verifies a simple condition
relative to the action of G [22, 8.1], we obtain a measurable family of representations
of G. In particular, if Z = G, the "tautological" family n f--> H1f yields a measurable
family.
If we are given a measurable family He, (( E Z) of representations of G, and a
positive measureμ on Z, we can form the representationon the space H = fz He, dμ(z) (natural definitions; [22, §8]).
Finally, assume n f--> m(n) is a Borel function from G to {O, 1, 2, 3, ... oo }. The
~ -m(7r)
map G 3 n f--> EJ1 H7r is clearly still a measurable bundle.
With these reminders we can now state the - essentially simple - properties
of the representation theory of G. As in the case of a finite or compact group, a
representation of G has an "essentially unique" decomposition into irreducibles.
Theorem 3.3. -Assume p is a unitary representation of G on a Hilbert space H.
(i) There exist a Borel function n f--> m(n) from G to {l, 2, ... oo} and a
positive measure μ on G such that
(3.6)
(ii) This expression is essentially unique. Precisely, if p has two expressions
(3.6), the measuresμ andμ' are equivalent and the functions m and m' are equal
almost everywhere.(^3) This Borel structure is standard, i.e., Gisin bijection with [O, l] endowed with the usual Borel
structure [22, 4.6.l].