A function is a utility function if it satisfies: u:X -> R x >= y <=> u(x) => u(y)

uii=u(xii)

Existence if => a weak order:
Cantor's theorem in 1895 says:
=> a weak order on X /\ X/~ (the equivalence classes) at most numerable => there exists u:X->R s.t. forall x,y in X x=>y <=> u(x)=>u(y)

this is later extended to:
=> a weak order on X /\ X/~ (the equivalence classes) has a dense numerable subset=> there exists u:X->R s.t. forall x,y in X x=>y <=> u(x)=>u(y)

Utility can be measured with a randomizing device. let w in O = states of the world
X= a set of consequences
F= a set of acts f:O->X

Let O be finite => 2^O is an algebra.

Find a probability charge P on (O,2^O)
Find a utility function u:X->R s.t.
f=>g <=> Sum(w in O,u(f(w))p(w))=>Sum(w in O,u(g(w))p(w))

The construction:
There exists E Subset of O which is ethically neutral.

There exists x">"y,x'">"y'

(x,E;y,Ec)=>(x',E;y',Ec) <=> u(x)-u(x')=>u(y)-u(y')

Method:
find the smallest and largest X (xl,xu) , assign them u(x)=0,1 respectively. let u(xl,E;xu,Ec)=1/2... build in from there.