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<b<=1 => 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.