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This book is intended to be an exhaustive study on regularity and other properties of continuity for different types of non-additive multimeasures and with respect to different types of topologies. The book is addressed to graduate and postgraduate students, teachers and all researchers in mathematics, but not only. Since the notions and results offered by this book are closely related to various notions of the theory of probability, this book will be useful to a wider category of readers, using multivalued analysis techniques in areas such as control theory and optimization, economic mathematics, game theory, decision theory, etc.
Measure and integration theory developed during the early years of the 20th century is one of the most important contributions to modern mathematical analysis, with important applications in many fields. In the last years, many classical problems from measure theory have been treated in the non-additive case and also extended in the set-valued case. The property of regularity is involved in many results of mathematical analysis, due to its applications in probability theory, stochastic processes, optimal control problems, dynamical systems, Markov chains, potential theory etc.
Introduction
References
Types of Multimeasures
References
Regular Multimeasures in the Hausdorff Metric Topology
Basic Notions and Results
Extensions of Regular Multimeasures
Set-Valued Egoroff and Lusin Type Theorems
Regular Non-atomic Multimeasures
References
Regular Multimeasures in the Vietoris Topology
Introduction
Basic Notions and Results
References
Approximation Theorems for Multimeasures in the Vietoris Topology
Introduction
Terminology, Basic Notions and Results
Convergences in the Vietoris Topology
Set-Valued Egoroff and Lusin Type Theorems in the Vietoris Topology
Certain Physical Implications and Correlations
Regularity of the Multimeasures in the Wijsman Topology
References
Regularity and Atomicity in an Abstract Hypertopology for Multimeasures
Introduction
Basic Notions and Terminology
Considerations on Non-atomicity and Regularity
Regularity in Its Most General Sense
References
Regularity of the Gould and Choquet Integrals
Introduction
Preliminaries
Monotone Multimeasures (With Respect to an Arbitrary Order Relation)
Gould Integrability
Important Properties of the Gould Integral
Certain Regularity Properties of the Gould and Choquet Interval-Valued Integrals
References
Multivalued Functions and Related Integrals
Introduction
Families of Subsets of Banach space
Measures
Fixed Point Theorems
Some Multivalued Additive Integrals
Multimeasures and Integrals
Multimeasures
Integration with Respect to Multimeasures
Index