4 - Directed index set

Summary

In this Chapter we present the theory of martingales and amarts indexed by directed sets. After Dieudonné showed that martingales indexed by directed sets in general need not converge essentially, Krickeberg—in a series of papers—proved essential convergence under covering conditions called “Vitali conditions.” This theory is presented in an expository article by Krickeberg & Pauc [1963] and in a book by Hayes & Pauc [1970].

Here we offer a new approach and describe the subsequent progress. The condition (V), introduced by Krickeberg to prove the essential convergence of L₁-bounded martingales, was shown not to be necessary. Similarly the condition (V^o), introduced to prove convergence of L₁-bounded submartingales, is now also known not to be necessary. The condition (V_Φ), which Krickeberg showed to be sufficient for the convergence of martingales bounded in the Orlicz space LΨ, is also necessary for this purpose if the Orlicz function ¸ satisfies the (Δ2) condition.

In each instance, the convergence of appropriate classes of amarts exactly characterizes the corresponding Vitali condition. This is of particular interest for (V) and (V^o) since there is no corresponding characterization in the classical theory. In general, to nearly every Vitali type of covering condition there corresponds the convergence of an appropriate class of “amarts.” The understanding of this fact was helped by new formulations of Vitali conditions in terms of stopping times. Informally, a Vitali condition says that the essential upper limit of a 0-1 valued process (1_At can be approximated by the process stopped by appropriate stopping times.