J . Zhang Mathematical Social Sciences 38 1999 11 –20
13
p: A →
[0,1] is a countably additive probability measure over if:
P.1 p550, pS 51; and P.2 p A 5
o pA , ;A A 55, for all i ±j.
n n
n n
i j
Denote by S, , p a l-system probability space. A probability p is convex-ranged if for all A[ and 0,r ,1, there exists B , A, B [, such that pB 5rpA. A
probability measure p is nonatomic if for all A[ with pA.0, there exists B , A, B [ such that 0,pB ,pA. It is well known that p is convex-ranged if and only if it
is nonatomic when is a s -algebra. However, it is an open question whether such equivalence is valid on a l-system.
2.2. Qualitative probabilities Let K be a binary relation . K is a qualitative probability if
Q.1 K is a weak order reflexive, complete and transitive; Q.2 AK5 for all A[;
Q.3 S s5; and Q.4 AC 5B C 55 implies [AKB
⇔ ACKB C].
A countably additive probability measure p on represents K if AKB
⇔ pA pB ,
for all A and B in .
`
The sequence hA j
in K-converges to A[ if for any events A , A in with
n n 51
A a Aa A, there exists an integer N such that A a A a A, whenever n N.
n
3. Main theorem
Given a qualitative probability K on a l-system , can we find a representing convex-ranged probability measure? Necessary and sufficient conditions are given next.
Denote by 15 5 A [ :A | 5 .
h j
n
A partition hA j
of S in is a uniform partition u. p if A | A | ? ? ? | A .
i i 51 1
2 n
Theorem 3.1. Let be a l-system and K a qualitative probability on . Then there
exists a convex-ranged , countably additive probability measure p on representing K
if and only if K satisfies the following :
14 J
. Zhang Mathematical Social Sciences 38 1999 11 –20
n
a i If A[\15, then there is a finite partition
hA j of S in such that
: 1
i i5
1 c
A , A or A , A , i51,2, . . . ,n; 2 A a A, i51,2, . . . ,n. ii If AaB and C A55,
i i
i m
then there is a finite partition hC j
of C in such that A C a B , i51,2, . . . ,m.
i i5
1 i
` `
b If hA j
is a decreasing sequence of events in , then
hA j K
-converges to
n n5
1 n
n5 1
`
A .
n5 1
n n
n
c For any two uniform partitions hA j
and hB j
of S in ,
A | B if
i i5
1 i
i5 1
i[ I i
i[ J i
uIu5uJu. Moreover
, under Conditions a –c, the representing measure p is unique. Conditions ai and aii are similar to Savage’s fine and tight ones, respectively.
Nonatomness is derived directly from Condition ai. Villegas 1964 employs a monotone continuity condition that applies both to increasing and decreasing sequences
of events. His condition is equivalent to our Condition b when is a s -algebra. That
`
is, if is a s -algebra, then a decreasing sequence hA j
K -converges to A if and
n n 51 n
c ` c
only if the increasing sequence hA j
K -converges to A . This is because
n n 51
n c
c
A a A ⇔
A s A . 3.1
However, the Eq. 3.1 is not true generally if is only a l-system. For example, consider a qualitative probability K on Eq. 1.1 as follows:
B,R,G,W s B,G s B,R s R,W s G,W s 5. h
j h j h
j h j h
j
c c
Then, hB,GjshB,Rj, but hB,Gj 5hR,WjshG,Wj5hB,Rj . This illustrates that Villegas’s
argument does not apply when the domain is only a l-system. Lemma 4.2 in Section 4 gives a sufficient condition for Eq. 3.1 on a l-system . The additional axiom c is
adopted here to compensate for the fact that is not a s -algebra. Obviously, Condition c follows from Q.4 if is an algebra.
4. Proof of Theorem 3.1
