Extremal problems with probability measures, functionally closed preorders and strong stochastic dominance

Utility Theory without the Completeness Axiom

“…We quote [14,9,13,20,4] where the existence of jointly continuous functions is proved in the case of linear preorders. Levin in [16,17,18] proved a general theorem for not necessarily linear preorders in second countable locally compact spaces. In the present paper we prove a generalization of Levin results in a nonmetrizable setting.…”

Mass Transportation Problems

“…The result was further extended to more general spaces by Szulga (1978Szulga ( , 1982, Fernique (1981), Huber (1981), Levin (1974Levin ( , 1984, Dudley (1976Dudley ( , 1989, de Acosta (1982), Rachev (1984b), Kellerer (1984b), Rachev and Shortt (1990) (see also Rachev (1991c, Chapter 6)). The result was further extended to more general spaces by Szulga (1978Szulga ( , 1982, Fernique (1981), Huber (1981), Levin (1974Levin ( , 1984, Dudley (1976Dudley ( , 1989, de Acosta (1982), Rachev (1984b), Kellerer (1984b), Rachev and Shortt (1990) (see also Rachev (1991c, Chapter 6)).…”

Utility representation theorems for Debreu separable preorders

“…In some cases the assumption ≼ to be closed, however, implies the validity of condition C5 or even the existence of some continuous utility function u : (X, ≼, t)  → (R, ≤, t nat ) (cf., for instance Bosi and Herden, 2006, Proposition 2.5, Herden and Pallack, 2002, Proposition 2.12, Levin, 1983, Theorems 1 and 2 and Levin, 1986, Theorems 3-6).…”

“…Following (Levin, 1986), we say that a preorder ≼ on (X, t) is functionally closed if it admits a representation ≼= {(x, y) | u(x) ≤ u(y) ∀u ∈ H} where H is a subset of the space C (X) of continuous real-valued functions on (X, t). It follows from this definition that a functionally closed preorder is closed.…”