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