Finitely additive 𝐹-processes

Armstrong, Thomas E.

doi:10.1090/S0002-9947-1983-0704616-9

Finitely additive $F$-processes
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Trans. Amer. Math. Soc. 279 (1983), 271-295 Request permission

Abstract:

If one replaces random variables by finitely additive measures one obtains instead of an $F$-process a finitely additive $F$-process. Finitely additive $F$-processes on a decreasing collection of Boolean algebras form a dual base norm ordered Banach space. When the collection is linearly ordered they form a dual Kakutani $L$-space. This $L$-space may be represented as the $L$-space of all finitely additive bounded measures on the Boolean ring of predictable subsets of the extreme points of the positive face of the unit ball. Of independent interest is the fact that any bounded supermartingale is a decreasing process in contrast to the usual case where only the supermartingales of class $DL$ are decreasing processes.

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