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')