On the Characters and the Plancherel Formula of Nilpotent Groups ...

298 DUDLEY
and the assignment
is well-defined.
WI * Ilc> = W) (9
By f.d.p. (finite-dimensional projection) we shall mean an ortho-
gonal projection of H onto a finite-dimensional subspace. For pro-
jections P and Q, one says P < Q if the range of P is included in that
of Q, and P, 7 I if PI < P2 < *** and P,(f) +f in (Hilbert) norm
for each f in H. Also P 1 Q means the ranges of P and Q are ortho-
gonal. If {fn} is an orthonormal basis of H, g, are independent,
normalized Gaussian random variables, and L,(f) = (f, f,) g, , then
the series
Ii1 Ln(9 (1’)
is a version of L. If P, 7 I (and the P, are f.d.p.‘s) then there is an
orthonormal basis { fi} of H such that for each n, { fi ,..., fk.} is a basis
of the range of P, for some k, , k, t co. The convergence of L o P,
to L is equivalent to convergence of a certain sequence of partial
sums of (1’).
We shall need an infinite-dimensional form of Proposition 4.0.
PROPOSITION 4.1. Let A be a linear operator from H into itself with
([AI/, Pr (L(C) < t).
Proof. If C is finite, the result follows immediately from Proposi-
tion 4.0. In general, let C, be finite sets which increase up to a dense
set in C. Then
Pr (L(C) < t) = $i Pr (L(C,) < t) < $i Pr (L(AC,) < t)
= Pr (L(AC) < t), Q.E.D.
PROPOSITION 4.2. If P,, are f.d.p.‘s, P, t I, C C H, and t >, 0,
then
Pr (L(C) < t) = $rr Pr (E(P,C) < t).
Proof. Let A be countable and dense in C. Then L(P,f) -L(f)
as n -+ co for all f in A, with probability 1. Hence
Pr (X(C) < t) < li?AdPr (yP,C) < t).