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This textbook is an introduction to rigorous probability theory using measure theory. It provides rigorous, complete proofs of all the essential introductory mathematical results of probability theory and measure theory. More advanced or specialised areas are only hinted at. For example, the text includes a complete proof of the classical central limit theorem, including the necessary continuity theorem for characteristic functions, but the more general Lindeberg central limit theorem is only outlined and is not proved. Similarly, all necessary facts from measure theory are proved before they are used, but more abstract or advanced measure theory results are not included. Furthermore, measure theory is discussed as much as possible purely in terms of probability, as opposed to being treated as a separate subject which must be mastered before probability theory can be understood. --This text refers to the Hardcover edition.
Preface
1 The need for measure theory
1.1 Various kinds of random variables
1.2 The uniform distribution and non-measurable sets
1.3 Additional exercises
1.4 Section summary
2 Probability triples
2.1 Basic definition
2.2 Discrete probability spaces
2.3 Constructing Lebesgue measure
2.4 The extension theorem
2.5 More on Lebesgue measure
2.6 Coin tossing and other measures
2.7 Additional exercises
2.8 Section summary
3 Further probabilistic foundations
3.1 Random variables
3.2 Independence
3.3 Continuity of probabilities
3.4 Limit events
A First Look at Rigorous Probability Theory, 2ed (Jeffrey S. Rosenthal).pdf
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