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

304 DUDLEY
If 11 * I/ is a measurable pseudo-norm on H*, then L is defined by a
countably additive probability measure on the completion of H*
for I/ * 11.l At the moment the converse seems to be an open question.
Suppose (xt , t E S) is a sample-continuous Gaussian process over a
compact metric space (S, d). Then t -+ xt is continuous from S into
H, and
e(s, t) = (E(xs - xJ2)l12
defines a pseudo-metric e on S which is continuous for d and hence
defines a weaker topology. If (S, e) is Hausdorff, i.e., if x, # xt
for s # t, then the d and e topologies on S are equal, and hence the
range of the process in H is a GC-set; its closed convex symmetric
hull is a GB-set, which, by Theorem 4.7, is not much worse.
5. SEQUENCES OF VOLUMES
We shall need the volumes of certain simple sets in Rk. First,
suppose A is a simplex, i.e., a convex hull of (h + 1) points x,, ,..., xk ,
having an interior. Let F be a face of A, i.e., a convex hull of K of its
vertices. Let h or h, be Lebesgue measure on Rk and S or Sk-r the
(K - 1)-dimensional Lebesgue “surface” or “area” measure. Then
h(A) = S(F) d/k,
where d is the distance from the vertex not in F to the hyperplane
through F. Now suppose x0 = 0 and let di be the distance from xj
to the linear span of x0 ,..., xi-r , j = l,..., k. Then
h(A) = (fJ dj)/k!.
i=l
Now, recalling the definitions of V,(C) and EV(C) given is Section
1, we have the following fact (a stronger statement is given as Propo-
sition 5.10 below):
LEMMA 5.0. If C is a convex set in H and EV(C) < - 1, then C
is totally bounded.
1 For the proof, see L. Gross, Abstract Wiener spaces, in Proceedings of the Fifth
Berkeley Symposium cm Mathematical Statistics and Probability (1964). University of
California Press, Berkeley, 1967.