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

294 DUDLEY
Proof. Given a positive integer n, we decompose S into N(1/2n+“)
sets of diameter at most l/2”+“, and choose one point from each set,
forming a set A, . Let G, be the set of all random variables L(x - y)
for x and y in A,-, u A, and jl x - y (1 < l/2%. Then the cardinality
of G, is at most 4N(2-n-4)2.
We shall use below the well-known estimate, for a > 0,
i
m 00 xe-“=I= dx e-a2/2
e-z212 dx & =-*
a I a a a
Let (b,) be any sequence of positive real numbers. Let
P, = Pr (max {l(z) 1 : z E G,} $ b,) < 4N (2-n-4)2 (exp [- 4%,2/2])2”/b,.
Thus P, < b, if n > 2 and - 4”bm2/2 + 2H(2-n-4) < 2 log b, ,
or
[H(2-“-4) - log !&J/4”-” < &a.
Let an2 = H(2-“-4)/4n-1. Then 2 a, < co by (l), and a, is inde-
pendent of b, . But now we specify b, , letting b, = max (2a, , l/n”).
Then an2 < bm2/2 and log b, > - 2 log n, so for n large enough
(- 4 log &J/4” < 1/2n4 < bn2/2,
and then P, < b, . Since C b, < co, we have 2 P, < co.
Thus for almost all w there is an no(w) such that 1 z 1 < b, for all
n > no(o) and all x in G, .
Now let T be any countable dense subset of 5’. We shall show that
on T, L is uniformly continuous with probability one. Its extension
to S is then a version of L with continuous sample functions, as
desired.
Given 6 > 0, we choose n, so that
where
sa(fkJ = +J : no(w) > no)
For any s in T, we choose points A,(s) in A, such that
II S - An(s) II < l/2 n+3. Now if n > n, , s, t E A, and (1 s -
then 11 A,(s) - A,(t) (1 f l/2%. Thus L(A,(s) - A,(t))
t 11 < 1/2n+3,
E G, . Also,
WW - An+,(s)) E Gn,, . Thus for w $ Q(n,,), L(A,(s)) (w) +L(s) (w)