By Mark J. Beran

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Then there exists a X, 0 < X < 1, such that 1 = ((1 - X)x + Xx) € N(x) f\C. 13). Hence 8{x) £ <*(x), which proves Property 3. THEOREM 1. i(x), • • • , gn{x) be differentiable functions on l'Jm. Let C be a convex set in Em and 8{x) be pseudo-convex on C and ffi(x), " • • » 9n(x) be quasi-convex on C. If there exist an x° € C and y" 6 E" satisfys A local minimum is atii E C such that 0(f) ^ 6{x) for all x (z N (x) M C, where N(x) is some neighborhood of x. 26 PART I. 14) V,0(x°) + V - .

7. Proof of Theorem 6. Let A l - 6 hold and let A be a denumerable partition of S wth P* (A) > 0 for all A e A. [P* (A )| A £ A] must have a largest element, say P (Ai). ). Continuing this process we get a sequence Ai, At, ••• with {Ai, A 2 , • • • j = A and (21) P*(Ai) g P*(A2) ^P*(A3) 6 ••• ; P*(A;)>0 for all i. For definiteness suppose that u is unbounded above. Let P , £ (P and P t 3C be such that (22) E(u,Pi) = P*(AiTl P = Pi on i - 1 , 2 , ••• A, t = 1,2, ••• . _„+i Ai. Letting v on 3C be as given in (8) and (9) and satisfying (10) on 3Co, (24) » ( 0 .

Introduction. The purpose of this paper is to present a general theory for the usual subjective expected utility model for decision under uncertainty. With a set S of states of the world and a set X of consequences let F be a set of functions on S to X. F is the set of acts. Under a set of axioms based on extraneous measurement probabilities, a device that is used by Rubin [14], Chernoff [3], Luce and Raiffa [9, Ch. 13], Anscombe and Aumann [1], Pratt, Raiffa, and Schlaifer [11], Arrow [2], and Fishburn [5], we shall prove that there is a realvalued function « o n I and a finitely-additive probability measure P on the set of all subsets of S such that, for all /, g e F, (1) / < S i f a n d o n l y i f £ [ u ( / ( s ) ) , P * ] g E[u(g(s)), P% In (1), < ("is not preferred to") is the decision-maker's binary preferenceindifference relation and E(y, z) is the mathematical expectation of y with respect to the probability measure z.

### Statistical continuum theories by Mark J. Beran

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