Ebook: 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.

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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.

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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.
Content:
Front Matter....Pages i-xiii
Sets and fuzzy sets....Pages 1-14
Null sets and small systems....Pages 15-33
Measures on ordered spaces....Pages 34-58
Subadditive measures....Pages 59-69
The Kurzweil integral in ordered spaces....Pages 70-102
Quantum logics....Pages 103-126
Fuzzy-quantum spaces....Pages 127-141
Fuzzy quantum logics....Pages 142-182
Probability on MV algebras....Pages 183-212
The entropy of fuzzy dynamical systems....Pages 213-251
Measurability and integrability of multifunctions....Pages 252-277
Back Matter....Pages 278-377

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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.
Content:
Front Matter....Pages i-xiii
Sets and fuzzy sets....Pages 1-14
Null sets and small systems....Pages 15-33
Measures on ordered spaces....Pages 34-58
Subadditive measures....Pages 59-69
The Kurzweil integral in ordered spaces....Pages 70-102
Quantum logics....Pages 103-126
Fuzzy-quantum spaces....Pages 127-141
Fuzzy quantum logics....Pages 142-182
Probability on MV algebras....Pages 183-212
The entropy of fuzzy dynamical systems....Pages 213-251
Measurability and integrability of multifunctions....Pages 252-277
Back Matter....Pages 278-377
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