Characterizations of Vitali Conditions with Overlap in Terms of Convergence of Classes of Amarts

Millet, Annie ; Sucheston, Louis

Canadian journal of mathematics, Tome 31 (1979) no. 5, pp. 1033-1046

Résumé

In a series of fundamental papers [20], [21], [22], [23], K. Krickeberg introduced 'Vitali’ conditions on σ-algebras and showed that they are sufficient for convergence of properly bounded martingales, and supermartingales. It is now known that the conditions V∞ (= V), and V′ are both sufficient and necessary for convergence of L1-bounded amarts, and ordered amarts (Astbury [1]; [24], [25]); an amart (ordered amart) is a process (Xt ) such that the net (EXτ)τ∈T* converges, where T* is the net of simple (ordered) stopping times. We undertake here to similarly characterize the Vitali conditions Vp, 1 ≦ p < ∞, in terms of convergence of properly defined classes of amarts. (In terms of convergence of L ∞-bounded martingales, Krickeberg himself [22] was able to characterize V 1.)

Détail

DOI : 10.4153/CJM-1979-095-2

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