Ps is the set of all simple charges on (X,A).
axioms for X finite:
* J1: "=>" a weak order on Ps
* J2: Forall p,q,r in Ps and a in [0,1] p=>q => ap+(1-a)r=>aq+(1-a)r (called independence or substitution)
* J3: forall p,q,r in Ps, if p>q>r => there exists a,b in [0,1] s.t. ap+(1-a)r>q>bp+(1-b)r (archimedean axiom) This axiom says there is nothing incomparably good or incomparably bad.
You can identify X with a subset of Ps: {p in Ps: p(x)=1 there exists x in X}=delta(x).
A representation theorem: (Von Neumann and Morgenstern)
Ps defined as above => "=>" on Ps satsifies J1,J2,J3 <=> there exists u:X->R s.t. forall p,q in Ps:
* p"=>"q <=> Sum(x in X, u(x)p(x))=>Sum(x in X, u(x)q(x))
with u unique up to positive affine transformation.
Note that:
* u(ap+(1-a)q)=au(p)+(1-a)u(q) with u(p)=Ep(u)
* axiom J3 removes the need to require "=>" a dense subset on Ps.
Proof:
Lemma: J1,J2,J3 hold =>
* p">"q and 0<=a** bp+(1-b)q>ap+(1-a)q
* p"=>"q"=>"r /\ p">"r => there exists unique a in [0,1] s.t. q~ap+(1-a)r
* p~q /\ a in [0,1] => p~q => ap+(1-a)r ~ aq+(1-a)r
Lemma: "=>" satisfies J1,J2,J3 => there exists xl,xu in X s.t. forall p in Ps,
delta(xu)=>p=>delta(xl)
Proof follows from above lemmas.**