STOCHASTIC

1424 PETER C. FISHBURN
delete the partition-specific subscript on u and have, in place of (5) when
P = Pi and Q = Qi on B,,
(6) P 4 Q if and only if £?_,£(«, Pt)PB*(Bi) § ~£UE(u, Qi)PB*(Bi).
For an event A C S let
3CX = {P | P £ X and P is constant on A and on A c \.
XA is a mixture set. Suppose each of partitions [B\, • • • , B„\ and (&,•••, Cm\
contains A. Then, if P in 3C^ equals PA on A and P/ on A", (6) implies that,
for all P, Q e XA ,
iV (.!)£(«, P4) + [1 - Pa* (A )]£(«, P/)
if and only if
Pc*(A)E(u, PA) + {1 - Pc*(A)]E(u, PA C )
g iV(A)£(u, QJ + [1 - PB* (A )}E(u, Q/)
S Pc*(A)E(u, QA) + [1 - PC*G4)]£(M, Q/).
It then follows easily from Theorem 1 that PB (A) = Pc (A). Hence we can
drop the partition-specific subscript on P and rewrite (6) as
(7) P 4 Q if and only if £?->#(«, Pi)P*(Bt) S T,UE(u, Qi)P*(Bi).
It follows directly from Theorem 2 that P is uniquely determined, that
P (A) = 0 only if A is null, and that u is unique up to a positive linear transformation.
Finite additivity for P is easily demonstrated using partitions
[A, B, (A u B)'\ and {A u B, (A u B)'} in an analysis like that leading to (7)
with inJ5 = 0.
Finally, to obtain (2) for all P, Q £ 3C0, let P = Pi on B{ and Q = Q, on Cj
for the partitions (Bi, • • • , £„} and {Ci, • • • , Cmj. Applying (7) to the partition
we obtain
\BinCj\i = 1, •••,«;.;' = 1, • • • , m; Bi n C3 # 0!
P < Q if and only if £, X^-Efa, Pi)P*(Bi n C,)
which, by finite additivity for P*, is the same as
S E(«/> + (1 - a)Q) = av(P) + (1 - a)v(Q).
16 PART I. MATHEMATICAL TOOLS