Ergodic Theorems

Every one of the important strong limit theorems that we have seen thus far – the strong law of large numbers, the martingale convergence theorem, and the ergodic theorem – has relied in a crucial way on a maximal inequality. This is no accident: it can in fact be shown that a maximal inequality is a necessary condition for an almost everywhere convergence theorem. We will refrain from carrying out this program, since it is tangential to our main interests. Instead, in the sections below we will develop a systematic method for using maximal inequalities to deduce almost everywhere convergence theorems. Our primary focus will be on ergodic theorems; however, since the ergodic theorem is in certain ways linked to differentiation theory, we will begin by proving Lebesgue’s differentiation theorem. The strategy is simple to describe, if not always to implement. First, find a dense subspace of L1 for which the convergence theorem is easily established. Second, prove a maximal inequality (technically, a weak type-(1,1) inequality). Finally, use the maximal inequality and an approximation argument to deduce the convergence theorem for all of L1.