Schaffler S. Mathematics of Information. Theory and Applications...2024

Textbook in PDF format

Starting with the Shannon-Wiener approach to mathematical information theory, allowing a mathematical "measurement" of an amount of information, the book begins by defining the terms message and information and axiomatically assigning an amount of information to a probability. The second part explores countable probability spaces, leading to the definition of Shannon entropy based on the average amount of information; three classical applications of Shannon entropy in statistical physics, mathematical statistics, and communication engineering are presented, along with an initial glimpse into the field of quantum information. The third part is dedicated to general probability spaces, focusing on the information-theoretical analysis of dynamic systems.
The book builds on bachelor-level knowledge and is primarily intended for mathematicians and computer scientists, placing a strong emphasis on rigorous proofs.
Introduction
Symbols
List of Figures
List of Tables
Basics
Message and Information
Information and Chance
Countable Systems
The Entropy
The Maximum Entropy Principle
Conditional Probabilities
Quantum Information
General Systems
The Entropy of Partitions
Stationary Information Sources
Density Functions and Entropy
Conditional Expectations
References