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

SIZES OF COMPACT SUBSETS OF HILBERT SPACE 295
for all s in T, and for any t in T such that d(s, t) < 1/2no+a, we have
I L(s) (w) -L(t) (w> I < 8.
Letting 6 4 0, we see that L is uniformly continuous on T with
probability 1. Q.E.D.
COROLLARY 3.2. If S is a subset of a Hilbert space and r(S) < 2,
then S is a GC-set.
There are numerous examples of sets S with 7(s) = 2 which are
neither GC- nor GB-sets; see, e.g., Section 6 below. Moreover,
Theorem 7.1 below and its partial converse, due to Fernique [7],
indicate that, even when specialized to stochastic processes on the real
line, Theorem 3.1 is essentially the best possible result of its kind.
However, we prove in Proposition 6.10(a) below that no sufficient
condition for the GC-property of a Banach ball in terms of H(S, E)
is necessary, i.e., the GC-property is not a “monotone function” of
the function E + H(S, l ) as E J 0. Yet I make
Conjecture 3.3. If S is a GB-set (and hence if S is a compact
GC-set), then r(S) < 2.
In Sections 6 and 7 below, Conjecture 3.3 is proved in a number
of special cases. In the general case, I shall prove at present only the
following:
PROPOSITION 3.4. If S is a GB-set then S is totally bounded
(i.e., its closure is compact).
Proof. If S is a GB-set, it is certainly bounded. Suppose it is not
totally bounded. Then for some E > 0 there is an infinite sequence
{ fj}& in S such that the distance of fi+l from the linear span Fi of
fi ,..., fi is at least E for all j. Let
where ]I g, I/ >, E and g, 1. F, . Given M > 0, let
Then
A,={w:max(lL(f,)I:l~j~n} WI
= Pr W5J > M) Pr &J/2.