A probability charge, P:A->R, with A an algebra on the set X satisfies:

• P(empty set) = 0, P(X)=1
• forall C,B in A: B interset C = empty set => P(C union B) = P(C) + P(B)

Properties:

• 1=>p(B)=>0 forall B in A
• forall B,C in A, B subset of C => P(A) <=P(B)
• forall B,C in A, P(B untion C) = P(B) + P(C) - P(B intersect C)
• if Ai i=1,..,n s.t. Ai intersect Aj = empty set forall i,j => P(Union(i=1,n,Ai)) = Sum(i=1,n,P(Ai))

given P with S1,S2,S3 holding => P is a mixture space.

A probability charge can be measured with a randomizing device. First, measure a utility function with the randomizing device.

let x">"y">"z with y ~ (x,A;z,Ac) for some A subset of O. Define:
P(A)=(u(y)-u(z))/(u(x)-u(z))

you can also measure conditional probability.

if (x,A;y,Ac) ~ ((x',B;z,Bc),A;y,Ac) with u(x)!=0 and P(A)>0

=> P(B|A)=(u(x)-u(z))/(u(x')-u(z))
These definitions obey:

• P(A)+P(Ac)=1
• P(A|B)+P(Ac|B)=1
• P(A int B)=P(A)P(B|A)
• P(A int B)+P(A int Bc)=P(A)
• A int B = empty set => P(A union B)=P(A)+P(B)

=> (x,A;y,Ac)=>(x',B;y',Bc) <=> P(A)u(x)+P(Ax)u(y) => P(B)u(x')+P(Bc)u(y')