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

300 DUDLEY
(a) is equivalent to (e) since a linear functional on a normed space is
continuous if and only if it is bounded on the unit ball. The proof
is complete.
Before treating GC-sets, we introduce some facts we need about
function-valued random variables. Let S be a metric space with a
countable dense subset A = {+‘ZZ1 . Let %(S) be the Banach space
of bounded continuous real-valued functions on S, with supremum
norm II * IL . We say Xi, X, ,... are given as a set of V(S)-valued
random variables if probabilities
Pr (Xi(ti) E Aij , i, j = 1, 2 ,...)
are defined for any points t, , t, ,..., in S and Bore1 sets A, in the
real line. Then the norms
II Xi I/m = SUP {I xi(t) I : t E A)
are measurable. Note, however, that W(S) will not be separable if S
is not compact. Then, the distributions of the Xi will not be expected
to be defined on all open sets in V(S) for the supremum norm topo-
lo!3 (cf. PI).
A random variable X in V(S) will be called symmetric if - X has
the same distribution as X. Independence of random variables Xi
in q(S) is defined also, naturally, to mean that the sets of real random
variables
Ai = {Xi(t) : t E S}
are independent for different values of i.
Let X, be independent and symmetric in %‘(S) and
The following generalization of a Lemma of P. Levy is proved much
like the classical version (Loeve [23], p. 247).
LEMMA 4.4. For any LY > 0,
Pr (max {\I S, 11 : k = l,..., 4 > 4 < 2 Pr (II S, II > 4.
Proof. For each k = l,..., m, j = 1, 2 ,..., and s = f 1, let
Afk, j, s) = {w : )I Si 11 < a, i = l,..., k - 1, ) &(x0) 1 < OL,
q = l,..., j - 1, sS,&) > a}.