1. SECTION 1 gives some examples of martingales, submartingales, and supermartingales.

2. SECTION 2 introduces stopping times and the sigma-fields corresponding to “information available at a random time.” A most important Stopping Time Lemma is proved, extending the martingale properties to processes evaluted at stopping times.

3. SECTION 3 shows that positive supermartingales converge almost surely.

4. SECTION 4 presents a condition under which a submartingale can be written as a difference between a positive martingale and a positive supermartingale (the Krickeberg decomposition). A limit theorem for submartingales then follows.

5. SECTION *5 proves the Krickeberg decomposition.

6. SECTION *6 defines uniform integrability and shows how uniformly integrable martingales are particularly well behaved.

7. SECTION *7 show that martingale theory works just as well when time is reversed.

8. SECTION *8 uses reverse martingale theory to study exchangeable probability measures on infinite product spaces. The de Finetti representation and the Hewitt-Savage zero-one law are proved.

What are they?

The theory of martingales (and submartingales and supermartingales and other related concepts) has had a profound effect on modern probability theory. Whole branches of probability, such as stochastic calculus, rest on martingale foundations. The theory is elegant and powerful: amazing consequences flow from an innocuous assumption regarding conditional expectations. Every serious user of probability needs to know at least the rudiments of martingale theory.

A little notation goes a long way in martingale theory. A fixed probability space (Ω, ℱ, ℙ) sits in the background.