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

Published online by Cambridge University Press:  20 November 2018

Annie Millet  and
Louis Sucheston
Annie Millet
Affiliation:
Ohio State University, Columbus, Ohio
Louis Sucheston
Affiliation:
Ohio State University, Columbus, Ohio
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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 V1.)

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 31 , Issue 5 , 01 October 1979 , pp. 1033 - 1046
Copyright
Copyright © Canadian Mathematical Society 1979

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