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.
Author(s): Alina Gavriluţ, Endre Pap
Series: Studies in Systems, Decision and Control, 448
Publisher: Springer
Year: 2022
Language: English
Pages: 165
City: Cham
Contents
1 Introduction
References
2 Types of Multimeasures
References
3 Regular Multimeasures in the Hausdorff Metric Topology
3.1 Basic Notions and Results
3.2 Extensions of Regular Multimeasures
3.3 Set-Valued Egoroff and Lusin Type Theorems
3.4 Regular Non-atomic Multimeasures
References
4 Regular Multimeasures in the Vietoris Topology
4.1 Introduction
4.2 Basic Notions and Results
References
5 Approximation Theorems for Multimeasures in the Vietoris Topology
5.1 Introduction
5.2 Terminology, Basic Notions and Results
5.3 Convergences in the Vietoris Topology
5.4 Set-Valued Egoroff and Lusin Type Theorems in the Vietoris Topology
5.5 Certain Physical Implications and Correlations
5.6 Regularity of the Multimeasures in the Wijsman Topology
References
6 Regularity and Atomicity in an Abstract Hypertopology for Multimeasures
6.1 Introduction
6.2 Basic Notions and Terminology
6.3 Considerations on Non-atomicity and Regularity
6.4 Regularity in Its Most General Sense
References
7 Regularity of the Gould and Choquet Integrals
7.1 Introduction
7.2 Preliminaries
7.3 Monotone Multimeasures (With Respect to an Arbitrary Order Relation)
7.4 Gould Integrability
7.5 Important Properties of the Gould Integral
7.6 Certain Regularity Properties of the Gould and Choquet Interval-Valued Integrals
References
Appendix A Multivalued Functions and Related Integrals
A.1 Introduction
A.2 Families of Subsets of Banach space
A.3 Measures
A.4 Fixed Point Theorems
A.5 Some Multivalued Additive Integrals
Appendix B Multimeasures and Integrals
B.1 Multimeasures
B.2 Integration with Respect to Multimeasures
Index
