Representation theorems typically say something about how a set of axioms can be turned into a utility function.

Ex: thm: "=>" on a finite X with completeness /\ transitivity <=> there exists

        u: X->R which represents "=>" i.e.  forall x,y in X: x=>y <=> 
        u(x)=>u(y).  u is unique up to monotonic increasing transformations.

variants include the <=> only being a =>. Also the utility function may be linear or additive.

Here are some representation theorems:
Theorem: X countable => "=>" a weak order <=> there exists u:X->R which represents "=>"

"=>" a waek order on X and u:X->R represents "=>" => v:X->R represents "=>" <=> there exists phi:R->R s.t

Note: I use ^ to denote inverse.
u /\ v represent "=>"
=> u(x)>u(y) <=> v(x)=v(y) /\ u(x)=u(y) <=> v(x)=v(y) for u and v on X/~. Suppose u(a)=c /\ u(b)=d => a=u^(b), b=u^(d). c>d <=> v(u^(c)) > v(u^(d)) => v(u^()) is strictly increasing with phi=v(u^()).

X subset of R^n => "=>" continuous weak order <=> there exists u:X->R a continuous representation of "=>". This theorem is generalizable to an arbitrary topology.

If a topology has a countable base => the above theorem with topology t instead of R^n

t=topology={empty set,(0,1)}=indiscreet topology The only continuous function in t are f(x)=constant forall x in (0,1). => there exists no representation of "=>" on topology t.

X=I1 x I2 x ... x In = cartesian product of intervals x => y <=> x !=y /\ xi=>yi forall i.


The extension "=>'" is representable by u:X->R =>

> a strict partial order on X /\ X/~~ countable => there exists u:X->R s.t. x>y => u(x)>u(y) /\ x~~y => u(x)=u(y).

If there exists u:P->R satisfying S1,S2,S3 /\ J4 => U on X defined by u(x)=U(delta(x)) is bounded.

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