Anna Martellotti

The Finitely Additive Integral of Multifunctions with Closed and Convex Values

We investigate integration with respect to a finitely additive measure of integrands with closed, convex values and we obtain a closedness result for the Aumann integral.

A minimax theorem for functions taking values in a Riesz space

On demontre des theoremes de minimax pour des fonctions prenant des valeurs dans un espace de Riesz. On donne une bonne extension de la version en espace de Riesz du theoreme de Hahn-Banach classique

Topological properties of the range of a group-valued finitely additive measure

INTRODUCTION In some recent works, various authors have extended many results already known in the literature and concerning the topological properties of the range of countably additive measures (c.a.m.‘s) to the case of finitely additive measures (f.a.m.‘s). In [18], the following theorem has been proved: the range of a con- tinuous f.a.m. whose target is R,, + is a closed interval. Therefore, in the case of R,f-valued f.a.m.3, everything works exactly as in the countably additive case. In [4] and [S], instead, the problem of establishing analogies and dif- ferences which arise in moving from the countably additive case to the finitely additive one when the set function takes its values in R” or, more generally, in a Banach space, has been investigated. In particular the authors showed that, while LiapounoII’s classical theorems have to be weakened in the finite-dimensional case (“the range of a continuous W- valued f.a.m. is convex though not compact in general”), in the inlinite- dimenional case all the classical results can be obtained again: I refer to [S J for the discussion and the related references. The aim of this paper is the investigation of the topological properties of the range of an f.a.m. with values in a topological group. In many recent papers [6, 10, 14, 191 several results have been obtained in the countably additive case. This paper tries to extend them to a weaker case. The procedures used in [S] for a Banach-space-valued f.a.m. are founded on a well-known Stone-type extension technique which enables us to generate a c.a.m. from a finitely additive one. Moreover such procedures make use of the density of the range of the f.a.m. in the range of its countably additive extension measure. Similar extension procedures may also be carried out in the case of topological group-valued set functions, but they are not as useful for the 411

Inequality systems and minimax results without linear structure

Inequality systems and their connections with minimax results for real- valued functions have been studied in different ways in several papers [ 1, 2,4, 5, 7-9, 133. In particular Pomerol [ 131 takes into consideration the equivalence between the consistency of inequality systems and the resolvability of each inequality of the system; it should be remarked that a minimax theorem can be directly derived from this equivalence. The aim of this paper is to investigate the relationships between the con- sistency of inequality systems and minimax results for functions defined on spaces without linear structure and taking values in a Riesz space; for real-valued functions, minimax results without linear structure go back to Fan [3]. Throughout this paper

The Burkill-Cesari Integral on Spaces of Absolutely Continuous Games

We prove that the Burkill-Cesari integral is a value on a subspace of and then discuss its continuity with respect to both the and the Lipschitz norm. We provide an example of value on a subspace of strictly containing as well as an existence result of a Lipschitz continuous value, different from Aumann and Shapley’s one, on a subspace of .

Coalitional Properness in Finitely Additive Economies

We define the notion of properness in the framework of coalitional economies. Through this, we obtain a core-Walras equivalence theorem for exchange economies with a measure space of agents and an infinite dimensional commodity space, whose positive cone has possibly empty interior.

Some Condition for Scalar and Vector Measure Games to Be Lipschitz

We provide a characterization for vector measure games in , with vector of nonatomic probability measures, analogous to the one of Tauman for games in , and also provide a necessary and sufficient condition for a particular class of vector measure games to belong to .

Fixed Point Theorems for Middle Point Linear Operators in

We introduce the notion of middle point linear operators. We prove a fixed point result for middle point linear operators in . We then present some examples and, as an application, we derive a Markov-Kakutani type fixed point result for commuting family of -nonexpansive and middle point linear operators in .

On the local weight theorem

It is widely known that weight and particularly weight loss is an important health topic and that its economical influence is of enormous importance nowadays ([8], [5], [6]). Indeed it is not presumptuous to claim that without weight business there would be many more depressed areas in the world and that most of the western economies would undergo a dramatic downfall: just to mention a few significant examples we might quote the future of firms like Sweet’n Low, of companies like Weight Watchers let alone Mr. Richard Simmons’ success. Therefore, with no doubt any mathematical model concerning weight loss should be labelled as applied mathematics and consequently be given full financial support. In this short note, following an idea originally due to Lavoisier, we state a theorem of Local conservation of weight, and derive from it a few of the several important consequences. Our Main Theorem can be interpreted