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

SIZES OF COMPACT SUBSETS OF HILBERT SPACE 299
On the other hand Proposition 4.1 yields
liy+yp Pr (L(P,C) < t) < Pr (YC) < t),
completing the proof.
The following definition is essentially that of Gross [9].
DEFINITION. A pseudo-norm 11 l I] on H* is measurable (for L) if
for every E > 0 there is a f.d.p. P, such that, for every f.d.p. P 1 PO ,
Pr (L(ll * II 0 P) > c) < E.
Note that ]I l 11 o P is a tame function on H* so that L of it is defined.
If C C H then I] * 11: is * measurable if and only if for every E > 0 there
is a f.d.p. P,, such that, for every f.d.p. P 1 PO ,
Pr @(PC) > c) < E.
It then follows by Propositions 4.1 and 4.2 that
Pr (EP,IC) > 6) < E.
(For any projection P, P 1 = I - P where I is the identity operator.)
THEOREM 4.3. If C is a Banach ball in H, the following are equiv-
alent:
(a) C is a GB-set, i.e., Pr (EC) < co) = 1.
(b) Pr (E(C) < co) > 0
(4 [resP* WI WnC) converges in law for some (resp. every)
sequence of f.d.p.‘s P, 1 I.
(e) L restricted to s(C) has a version linear and continuous with
probability I for /I * II=.
Proof. Let {f,J b e an orthonormal basis of H. For each f in H,
the series (1’) converges almost everywhere on 52 and in L2(Q). For
any finite N,
@(9 b)
is bounded on C for each w in Sz, and finiteness of L(C) (0) thus
depends on the g, for n > N. Thus by the zero-one law ([13], B,
p. 229), Pr (E(C) < co) = 0 or 1, and (a) is equivalent to (b).
(a) is equivalent to (c) and (d) by Proposition 4.2.