Positive Operators, Riesz Spaces, and Economics: Proceedings of a Conference at Caltech, Pasadena, California, April 16–20, 1990 (Studies in Economic Theory, 2)

The volume contains the proceedings of a conference in ma- thematical economics held at the California Institute of Technology, April 16-20, 1990.The theme of the conference was to explore the functional analytic techniques (and in particular the theory of Riesz spaces and positive opera- tors) that are employed in economics. The articles in the book depict this important connection and open the way for more economic applications. "Over the last fifty years advanced mathematical tools have become an integral part in the development of modern economic theory. Economists continue to invoke sophisticated mathematical techniques and ideas in order to understand complex economic and social problems. In the last ten years the theory of Riesz spaces (vector lattices) has been successfully applied to economic theory. By now it is understood relatively well that the lattice structure of Riesz spaces can be employed to capture and interpret several economic notions. On April 16-20, 1990, a small conference on Riesz Spaces, Positive Opera tors, and their Applications to Economics took place at the California Institute of Technology. The purpose of the conference was to bring mathematicians specialized in Riesz Spaces and economists specialized in General Equilibrium together to exchange ideas and advance the interdisciplinary cooperation between mathematicians and economists. This volume is a collection of papers that represent the talks and discussions of the participants at the week-long conference. We take this opportunity to thank all the participants of the conference, especially those whose articles are contained in this volume. We also greatly appreciate the financial support provided by the California Institute of Technology. In particular, we express our sincerest thanks to David Grether, John Ledyard, and David Wales for their support. Finally, we would like to thank Susan Davis, Victoria Mason, and Marge D'Elia who handled the delicate logistics for the smooth running of the conference." --Publisher Over the last fifty years advanced mathematical tools have become an integral part in the development of modern economic theory. Economists continue to invoke sophisticated mathematical techniques and ideas in order to understand complex economic and social problems. In the last ten years the theory of Riesz spaces (vector lattices) has been successfully applied to economic theory. By now it is understood relatively well that the lattice structure of Riesz spaces can be employed to capture and interpret several economic notions. On April 16-20, 1990, a small conference on Riesz Spaces, Positive Opera­ tors, and their Applications to Economics took place at the California Institute of Technology. The purpose of the conference was to bring mathematicians special­ ized in Riesz Spaces and economists specialized in General Equilibrium together to exchange ideas and advance the interdisciplinary cooperation between math­ ematicians and economists. This volume is a collection of papers that represent the talks and discussions of the participants at the week-long conference. We take this opportunity to thank all the participants of the conference, especially those whose articles are contained in this volume. We also greatly ap­ preciate the financial support provided by the California Institute of Technology. In particular, we express our sincerest thanks to David Grether, John Ledyard, and David Wales for their support. Finally, we would like to thank Susan Davis, Victoria Mason, and Marge D'Elia who handled the delicate logistics for the smooth running of the confer­ ence. Title......Page 3 Foreword......Page 5 Contents......Page 7 Valuation and Optimality in Exchange Economies with a Countable Number of Agents......Page 9 Equilibrium Points of Non-Cooperative Random and Bayesian Games......Page 31 Equilibria of Large Games with Imperfect Observability......Page 57 Functional Analytic Tools for Expected Utility Theory......Page 77 Remarkable Points and X(N)-spaces......Page 97 Integration with Respect to Finitely Additive Measures......Page 117 Lattice-Ordered Algebras and f-Algebras: A Survey......Page 159 Approximating Derivative Securities in f-Algebras......Page 179 Some Unpleasant Objects in a Non-separable Hilbert Space......Page 187 The Hopf Decomposition in Riesz Spaces......Page 197 Frobenius Decomposition of Positive Compact Operators......Page 203 Irreducible Positive Operators and Hyperinvariant Ideals......Page 233

The volume contains the proceedings of a conference in ma-
thematical economics held at the California Institute of Technology, April 16-20, 1990.The theme of the conference was to explore the functional analytic techniques (and in particular the theory of Riesz spaces and positive opera-
tors) that are employed in economics. The articles in the book depict this important connection and open the way for more economic applications.

In exchange economies, having a finite number of agents and a finite number of commodities, there are two celebrated welfare theorems.