References & Further Reading

References

1. R. Durrett, Probability: Theory and Examples, Cambridge University Press, 5th ed., 2019. [Link]

   The primary reference for this chapter. Chapters 1 and 4 cover measure-theoretic foundations and conditional expectation with full proofs.

2. P. Billingsley, Probability and Measure, Wiley, 3rd ed., 1995

   A classic text that bridges the gap between elementary probability and measure theory. Especially strong on the Borel sigma-algebra and convergence theorems.

3. H. L. Royden and P. M. Fitzpatrick, Real Analysis, Pearson, 4th ed., 2010

   The standard graduate real analysis text. Chapters 2-4 cover Lebesgue measure and integration; Chapter 18 covers the Radon-Nikodym theorem.

4. D. Williams, Probability with Martingales, Cambridge University Press, 1991

   An elegant and concise introduction to martingales and conditional expectation. Chapters 9-10 are directly relevant to Section 22.3.

5. J. A. Gubner, Probability and Random Processes for Electrical and Computer Engineers, Cambridge University Press, 2006

   The primary FSP course textbook. Chapter 3 covers distributions including mixed types; Chapter 9 covers conditional expectation.

6. H. V. Poor, An Introduction to Signal Detection and Estimation, Springer, 2nd ed., 1994

   Connects the Radon-Nikodym theorem to hypothesis testing and detection theory. Chapter 2 develops the likelihood ratio in the measure-theoretic framework.

7. A. N. Shiryaev, Probability, Springer, 2nd ed., 1996

   A comprehensive probability text with strong measure-theoretic foundations. Good treatment of martingales and the optional stopping theorem.

8. G. B. Folland, Real Analysis: Modern Techniques and Their Applications, Wiley, 2nd ed., 1999

   A thorough real analysis reference with excellent coverage of measure construction, the Radon-Nikodym theorem, and product measures.

9. D. Tse and P. Viswanath, Fundamentals of Wireless Communication, Cambridge University Press, 2005

   Used here for the connection between mixed distributions and outage capacity in fading channels.

10. G. Caire, Foundations of Stochastic Processes (FSP) — Lecture Notes, TU Berlin, 2024

    The course lecture notes that this book is based on. Chapter 22 extends the measure-theoretic foundations only briefly treated in the course.

Further Reading

• Kolmogorov's axiomatization and its impact

  A. N. Kolmogorov, *Grundbegriffe der Wahrscheinlichkeitsrechnung*, 1933 (English translation: *Foundations of the Theory of Probability*, Chelsea, 1956)

  The foundational document that built probability theory on measure theory. Short and readable.

• Martingale theory in depth

  D. Williams, *Probability with Martingales*, Cambridge, 1991, Chapters 11-14

  Covers martingale convergence, uniform integrability, and the optional stopping theorem with full proofs and illuminating examples.

• Detection theory for Gaussian processes

  H. V. Poor, *An Introduction to Signal Detection and Estimation*, Springer, Ch. 4

  Shows how the Radon-Nikodym derivative for Gaussian processes leads to the correlator detector and the Cameron-Martin formula.

• Measure-theoretic information theory

  R. M. Gray, *Entropy and Information Theory*, Springer, 2011

  Develops information-theoretic quantities (entropy, mutual information, divergence) in full measure-theoretic generality.

• Stochastic integration and Ito calculus

  B. Oksendal, *Stochastic Differential Equations*, 6th ed., Springer, 2003

  The natural next step after martingales: stochastic calculus requires the measure-theoretic framework developed here.