1. GENERAL REPRESENTATIONS OF Od ON A SEPARABLE HILBERT SPACE 5

by the computation

R(focr)(x)= ] T G(y)f(a(y)) = f(x) £ G (y) = f (x) R (1) (x).

y y

cr(y)=x cr(y)=x

We observe that R(l) is a positive function by (1.14) and it is /i-integrable, as can

be seen by using (1.13) on g = 1. If / is a positive bounded function on Q we have

from (1.15):

(1.16) »(foT) = n(R(foa)) = »(R(l).f).

Putting / equal to characteristic functions, the a-quasi-invariance (1.5) of \i is

immediate. •

As a final note on invariance, observe that (1.11) implies the condition

/i(£) = 0 = M (*(£)) = 0,

by the following reasoning. Assume (1.11) throughout. If

\I(G(E))

= 0, then

fi (o-iG (E)) = 0 for all i, but as E C |J. GIG (E) it follows that \i (E) — 0. Con-

versely, if ii (E) = 0, write E as E = \Ji oi (Ei), and then fi (o~i (Ei)) \i (E) = 0,

and hence, as a (E) = [ji o~i (Ei), we have \i (a (E)) = 0.

We now come to the main result in this chapter.

Theorem 1.2. For any nondegenerate representation Si i— • Si of Od on a sepa-

rable Hilbert space H, there exists a probability measure /i on ft which is Oi-quasi-

invariant for i = 1,... ,d (and thus a-quasi-invariant by Proposition 1.1), and a

measurable direct integral decomposition

(1.17) H = / H(x) dfi(x)

Jn

of H into Hilbert spaces H(x) such that the spaces H(x) are constant (have fixed

dimension) over a-orbits in Q, and there exists a measurable field ft 3 x i— • U (x)

of unitary operators such that if

(1.18) f = / Z(x)dii(x)

is a vector in H, then

(1.19) Si£= [ (S^)(x)dfi(x),

(i.2o) s;z = / (s;z){x)dn(x),

Jn

where

(1.21) (S£) (x) =

Xi

(x) p (x) U (x) £ (a (a;)),

(1.22) (S*0 (x) = p (at

(x))-1

U (at (x))* £ (* (x)).

Here

(1.23) P\x)=\

\ dfi(x)