Let (O,A) where O=state space and A = algebra on the state space.

A"=>"B => A more likely than B.

p() is a function that represents "=>":

A"=>"B <=> p(A)=>p(B)

Properties of p:

• normalized => p(empty set)=0 p(O)=1
• additive A intersect B = empty set => p(A union B)=p(A)+p(B)

P is unique <=> there exists n-1 Ai ~ Bi in A linearly independent.

Theorem:
A an algebra on O, "=>" a relation on A =>

• there exists P:O->R which represents "=>" <=> "=>" satisfies QP1, QP4
• "=>" satisfies QP1,QP2,QP3 /\ P(empty set)=0 /\ P(O)=1 => forall B in A: 1=>P(B)=>0
• "=>" satisfies QP1,QP2,QP3,QP4 => there exists P /\ there exists G:R^2->R s.t. C intersect B = empty set, P(A union B)=G(P(A),P(B))
  • G(X,Y)=G(Y,X),
  • G strictly increasing in x, G(x,p(empty set))=x,
  • G associative G(G(x,y),z)=G(x,G(y,z))

The function G sometimes reduces to addition.

Theorem: O finite, A in 2^O =>

• there exists p:O->R representing "=>" and additive <=> "=> complete and QP5
• "=>" satisfies QP2 also => P(empty set) = 0 /\ P(O)=1 /\ pi=P(wi)=>0, Sum(pi)=1