On the continuity of expected utility

2014

Abstract. Suppose that X and Y are Banach spaces and that the Banach space X⊗̂τY is their complete tensor product with respect to some tensor product topology τ. A uni-formly bounded X-valued function need not be integrable in X⊗̂τY with respect to a Y-valued measure, unless, say, X and Y are Hilbert spaces and τ is the Hilbert space tensor product topology, in which case Grothendieck’s theorem may be applied. In this paper, we take an index 1 ≤ p < ∞ and suppose that X and Y are Lp-spaces with τp the associated Lp-tensor product topology. An application of Orlicz’s lemma shows that not all uniformly bounded X-valued functions are integrable in X⊗̂τpY with respect to a Y-valued measure in the case 1 ≤ p < 2. For 2 < p < ∞, the negative result is equivalent to the fact that not all continuous linear maps from ℓ1 to ℓp are p-summing, which follows from a result of S. Kwapien.

Boletin De La Sociedad Matematica Mexicana, 2016

Using classical techniques related to the so-called Hardy-Vitali variation, we present the class of X-valued functions of bounded-variation in several variables, where (X, d, +) is a metric semigroup. We exhibit some of the main properties of this class; among them, we show that this class can be made into a normed space and present a counterpart of the renowned Riesz's Lemma for the case in which X = R with its usual metric.

2009

Function Spaces - Proceedings of the Sixth Conference, 2003

Let E be a Banach function space and X be an arbitrary Banach space. Denote by E(X) the Köthe-Bochner function space defined as the set of measurable functions f : Ω → X such that the nonnegative functions f X : Ω → [0, ∞) are in the lattice E. The notion of E-variation of a measure -which allows to recover the pvariation (for E = L p ), Φ-variation (for E = L Φ ) and the general notion introduced by Gresky and Uhl-is introduced. The space of measures of bounded E-variation V E (X) is then studied. It is shown, among other things and with some restriction of absolute continuity of the norms, that (E(X)) * = V E (X * ), that V E (X) can be identified with space of cone absolutely summing operators from E into X and that E(X) = V E (X) if and only if X has the RNP property.

Journal of Mathematical Analysis and Applications, 1998

Mathematics of Operations Research

We obtain several variants of the classic von Neumann–Morgenstern expected utility theorem with and without the completeness axiom in which the derived Bernoulli utility functions are Lipschitz. The prize space in these results is an arbitrary separable metric space, and the utility functions are allowed to be unbounded. The main ingredient of our results is a novel (behavioral) axiom on the underlying preference relations, which is satisfied by virtually all stochastic orders. The proof of the main representation theorem is built on the fact that the dual of the Kantorovich–Rubinstein space is (isometrically isomorphic to) the Banach space of Lipschitz functions that vanish at a fixed point. An application to the theory of nonexpected utility is also provided.

2010

Abstract. Suppose that X and Y are Banach spaces and that the Banach space X⊗̂τ Y is their complete tensor product with respect to some tensor product topology τ . A uniformly bounded X-valued function need not be integrable in X⊗̂τ Y with respect to a Y -valued measure, unless, say, X and Y are Hilbert spaces and τ is the Hilbert space tensor product topology, in which case Grothendieck’s theorem may be applied. In this paper, we take an index 1 ≤ p < ∞ and suppose that X and Y are L-spaces with τp the associated L-tensor product topology. An application of Orlicz’s lemma shows that not all uniformly bounded X-valued functions are integrable in X⊗̂τpY with respect to a Y -valued measure in the case 1 ≤ p < 2. For 2 < p < ∞, the negative result is equivalent to the fact that not all continuous linear maps from l to l are p-summing, which follows from a result of S. Kwapien.

Let S be a locally compact Hausdorff space and let us consider the space C 0( S, X ) of continuous functions vanishing at infinity, from S into the Banach space X . A theorem of I. Singer, settled for S compact, states that the topological dual C 0*( S, X ) is isometrically isomorphic to the Banach space r σ bv ( S, X *) of all regular vector measures of bounded variation on S , with values in the strong dual X *. Using the Riesz-Kakutani theorem and some routine topological arguments, we propose a constructive detailed proof which is, as far as we know, different from that supplied elsewhere.

2002

In order to apply the direct methods of the calculus of variations to this class of functionals, a first problem to be solved is the identification of qualitative conditions on the supremand f which imply the lower semicontinuity with respect to a convergence weak enough to provide the compactness in a large number of situations, say the weak* L∞ convergence. This was already solved by Barron and Liu in [3] where they showed that a functional of the form (1) is weakly* L∞ sequentially lower semicontinuous if and only if the function

2016

In the present work, we study the possibility to represent the pre-orders defined by the characteristic linear operators of financial (or economic) markets represented in the States Preference Model perspective by applying classical theorems of the existence of jointly continuous utilities. We conduct our exam by a strongly application-oriented mood. Indeed, by introducing the classic setting of Arrow-Debreu State Preference Model and its recent generalization to the sphere of Schwartz Linear Algebra in distribution spaces, we address the problem of extending some fundamental results of finite dimensional State Preference Decision Theory to a new case, characterized by a hard type infinite linear-topological dimensionality. In this case a representation theorem for submetrizable kω-space is applied.