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')