
THEORETICAL VALIDITY OF IMS 61
1 −α}. Since each S is closed, so is S
α
; it is also measurable by our assumptions
about the richness of the σ-algebra on U. The key observation is that Q(u) > 1 −α
iff u ∈ S
c
α
. Therefore, by continuity of P
U
from above, we get
P
U
{Q
S
(U ) > 1 −α} = P
U
(S
c
α
) = 1 −P
U
(S
α
) = 1 −lim P
U
(S),
where the limit is over all S decreasing to S
α
. By construction, each such S satisfies
P
U
(S) ≥1−α. So, finally, we get P
U
{Q
S
(U ) > 1−α}≤α and, since α is arbitrary,
we get Q
S
(U ) ≤
st
Unif(0,1), proving validity.
Remark 4.1. The particular random set constructed (4.15) in Theorem 4.1, whose
distribution we call the natural measure, is an essential ...