Abstract Methods in Information Theory

    Information Theory is studied from the following view points: (1) the theory of entropy as amount of information; (2) the mathematical structure of information sources (probability measures); and (3) the theory of information channels. Shannon entropy and Kolmogorov-Sinai entropy are defined and their basic properties are examined, where the latter entropy is extended to be a linear functional on a certain set of measures. Ergodic and mixing properties of stationary sources are studied as well as AMS (asymptotically mean stationary) sources. The main purpose of this book is to present information channels in the environment of real and functional analysis as well as probability theory. Ergodic channels are characterized in various manners. Mixing and AMS channels are also considered in detail with some illustrations. A few other aspects of information channels including measurability, approximation and noncommutative extensions, are also discussed.

Preface
ChapterⅠ.Entropy
1.1 The Shannon Entropy
1.2 Conditional expectations
1.3 The Kolmogorow-Sinai entropy
1.4 Algebraic models
1.5 Entropy functionals
1.6 Relative entropy and Kullback-Leibler information
ChapterⅡ.Information Sources
2.1 Alphabet message spaces and information souces
2.2 Ergodic theorems
2.3 Ergodic and mixing properties
2.4 AMS sources
2.5 Shannon-McMillan-Breiman theorem
2.6 Ergodic decompositions
2.7 Enropy functionals,revisited
Bibilographical notes
ChapterⅢ.Information Channels
3.1 Information channels
3.2 Channel operators
3.3 Mixing channels
3.4 Ergodic channels
3.5 AMS channels
3.6 Capacity and transmission rate
3.7 Coding Therorems
Bibliographical notes
ChapterⅣ.Special Topics
4.1 Channels with a noise source
……
Special Topics
Indices