Mixture-Space Theorem

The mixture-space theorem, proved by Israel Nathan Herstein and John Milnor in 1953, is a utility-representation theorem in microeconomic and decision theory showing that a preference relation defined over a general mixture space, one satisfying axioms of completeness, independence and mixture continuity, can be represented by a utility function that is linear with respect to the space's mixture structure. Herstein and Milnor's paper both introduced the formal definition of a mixture space and proved the theorem for it, generalizing both the von Neumann-Morgenstern expected-utility theorem and the standard utility-representation theorems used for ordinary consumer preferences.