1. This result is called the von Neumann Morgenstern utility representation theorem . 2. Existence and uniqueness of this operator follows from the Riesz representation theorem . 3. For the theorems relating linear functionals to measures, see Riesz Markov Kakutani representation theorem . 4. This is the Kolmogorov Arnold representation theorem . 5. The result is a distributive lattice and is used in Birkhoff's representation theorem . 6. The other direction is less trivial, in that it requires the representation theorems stated below. 7. The significance of positive linear functionals lies in results such as Riesz Markov Kakutani representation theorem . 8. So in that case " A " can be given by Riesz representation theorem . 9. Furthermore, viewing states as normalized functionals, and invoking the Riesz representation theorem , we can put 10. By the way, I am not sure at all that the representation theorem should be on wikipedia.
