Integral, Measure, and Ordering

The present book is a monograph including some recent results of mea sure and integration theory. It concerns three main ideas. The first idea deals with some ordering structures such as Riesz spaces and lattice or dered groups, and their relation to measure and integration theory. The second is the idea of fuzzy sets, quite new in general, and in measure theory particularly. The third area concerns some models of quantum mechanical systems. We study mainly models based on fuzzy set theory. Some recent results are systematically presented along with our suggestions for further development. The first chapter has an introductory character, where we present basic definitions and notations. Simultaneously, this chapter can be regarded as an elementary introduction to fuzzy set theory. Chapter 2 contains an original approach to the convergence of sequences of measurable functions. While the notion of a null set can be determined uniquely, the notion of a set of "small" measure has a fuzzy character. It is interesting that the notion of fuzzy set and the notion of a set of small measure (described mathematically by so-called small systems) were introduced independently at almost the same time. Although the axiomatic systems in both theories mentioned are quite different, we show that the notion of a small system can be considered from the point of view of fuzzy sets.

Klappentext

This book is concerned with three main themes. The first deals with ordering structures such as Riesz spaces and lattice ordered groups and their relation to measure and integration theory. The second is the idea of fuzzy sets, which is quite new, particularly in measure theory. The third subject is the construction of models of quantum mechanical systems, mainly based on fuzzy sets. In this way some recent results are systematically presented. Audience: This volume is suitable not only for specialists in measure and integration theory, ordered spaces, probability theory and ergodic theory, but also for students of theoretical and applied mathematics.

Inhalt
1 Sets and fuzzy sets.- 2 Null sets and small systems.- 3 Measures on ordered spaces.- 4 Subadditive measures.- 5 The Kurzweil integral in ordered spaces.- 6 Quantum logics.- 7 Fuzzy-quantum spaces.- 8 Fuzzy quantum logics.- 9 Probability on MV algebras.- 10 The entropy of fuzzy dynamical systems.- 11 Measurability and integrability of multifunctions.- Appendix A D-posets (Ferdinand Chovanec, Frantisek Kôpka).- A.1 Difference posets.- A.2 Difference lattices.- A.3 Compatibility in D-posets.- A.4 States and observables on D-posets.- A.5 Ideals in D-posets.- Notes and comments.- Appendix B Notes on order convergence and order topology (Hana Kirchheimová, Zdenka Rie?anová).- B.1 Order convergence of nets and filters in partially ordered sets and their MacNeille completions.- B.2 Order topology and order topological lattices in language of filters and nets.- B.3 Metric modular ortholattices.- Problems.- Notes and comments.- References.