This textbook presents measure theory in a concise yet clear manner, providing readers with a solid foundation in the mathematical axiomatic system of probability theory. Unlike elementary probability theory, which deals with random events through specific examples of random trials, Foundations of Probability Theory offers a comprehensive mathematical framework for rigorous descriptions of these events.

As a result, this course embodies all the characteristics of mathematical theories: abstract content, extensive applications, complete structures, and clear conclusions. Due to the abstract nature of the material, learners may encounter various challenges. To overcome these difficulties, it is essential to keep concrete examples in mind when trying to understand abstract concepts and to compare the abstract theory with related courses previously studied, particularly the Lebesgue measure theory.

To enhance the readability of the book, each section begins with a brief introduction outlining the main objectives based on the preceding content, highlighting the primary structure, and explaining the key ideas of the study. This approach ensures that readers can follow the material more easily and grasp the essential concepts effectively.

Contents:

• Class of Sets and Measure


• Random Variable and Measurable Function


• Integral and Expectation


• Product Measure Space


• Conditional Expectation and Conditional Probability


• Characteristic Function and Weak Convergence


• Probability Distances


• Calculus on the Space of Finite Measures

Readership: Undergraduate students, graduate students, researchers and university teachers.

Key Features:

• Measure theory, or foundation of probability theory, is the basis for the study of probability theory and applications


• Comparing to many other textbooks on this topic, the present book provides an easy to follow line for readers to understand the key idea of the theory, global structure of knowledge, and a clear picture of the system