STOCHASTIC

1420 PETER C. FISHBTJBN
place no special restrictions on S but they imply that X is infinite and that if
u(x) < u(y) then there is a z E X such that u(z) = .5u(x) + .5u(y). On the
other hand Savage [15] does not restrict X in any unusual way, but his theory
requires S to be infinite and implies that, for any positive integer n, there is an
n-part partition of S such that P = 1/n on each part of the partition. Arrow
[2] also assumes this property for P .
2. Definitions and notation. (P is the set of all simple probability measures
(gambles) on X, so that if P E (P then P(Y) = 1 for some finite Y included in
X. The probabilities used in (P are extraneous measurement probabilities. They
can be associated with outcomes of chance devices such as dice and roulette
wheels.
With P, Q e. Under this interpretation, (P is a
mixture set. By Herstein and Milnor's [7] definition, a mixture set is a set M and
an operation that assigns an element aa + (1 — a)b in M to (a, b) £ M x M
and a £ [0, 1] in such a way that
(l)o + (0)6 = a
aa + (1 — a)b = (1 — a)b + aa
a{0a + (1 - 0)6) + (1 - a)b = (a/3)a + (1 - a0)b
for all a,b EM and a, f$ E [0, 1].
3C is the set of all functions on S to (P. With P E 3C, P(S) is the gamble in