In this chapter, we will prove some of the pointwise convergence theorems from ergodic theory. The main result of this chapter is the superadditive ratio ergodic theorem. It implies the Chacon-Ornstein theorem, the Kingman subadditive ergodic theorem, and, for positive operators, the Dunford-Schwartz theorem and Chacon's ergodic theorem involving “admissible sequences.” We consider positive linear contractions T of L1.

Our plan is as follows. We first prove weak maximal inequalities, from which we obtain the Hopf decomposition of the space Ω into the conservative part C and the dissipative part D (8.3.1). Assuming T conservative (that is, Ω = C, we prove the Chacon-Ornstein theorem (8.5.4), i.e., the convergence to a finite limit of the ratio of sums of iterates of T applied to functions f and g The limit is identified in terms of f, g, and the σ-algebra C of absorbing sets. The superadditive operator ergodic theorem is proved in the conservative case (8.4.6). It is then observed that the total contribution of Ω to C is a superadditive process with respect to the conservative operator Tc induced by T on C. Since the behavior of the ergodic ratio on the dissipative part D is obvious, the Chacon-Ornstein theorem (8.6.10) and, more generally, the superadditive ratio theorem (8.6.7) will follow on Ω This affords considerable economy of argument, since the direct study of the contribution of D to C is not obvious even for additive processes.

The superadditive theory (or, equivalently, the subadditive theory) is mostly known for its applications, but in fact the notion of a superadditive process is shown to shed light on the earlier additive theory of L1 operators.