In this chapter a detailed introduction to martingale theory is presented. In particular, we study important Banach spaces of martingales with regard to the supremum norm and the quadratic variation norm. The main results show that the martingales in the associated dual spaces are of bounded mean oscillation. The Burkholder–Davis–Gandy (B–D–G) inequalities for Lp-bounded martingales are very useful applications. All results in this chapter are well known; I learned the proofs from Imkeller's lecture notes. We also need the B–D–G inequalities for special Orlicz spaces of martingales.

In this chapter we study martingales, defined on standard finite timelines. Later on the notion ‘finite’ is extended and the results in this chapter are transferred to a finite timeline, finite in the extended sense. We obtain all established results also for the new finite timeline. Then we shall outline some techniques to convert processes defined on this new finite timeline to processes defined on the continuous timeline [0,∞[ and vice versa. The reader is referred to the fundamental articles of Keisler, Hoover and Perkins and Lindstrøm.

From what we have now said it follows that we only need to study martingales defined on a discrete, even finite, timeline.