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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



Finitely additive $F$-processes

Author: Thomas E. Armstrong
Journal: Trans. Amer. Math. Soc. 279 (1983), 271-295
MSC: Primary 60G48; Secondary 28A60, 60G07, 60G40
MathSciNet review: 704616
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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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Keywords: Supermartingales, martingales, potentials, <IMG WIDTH="21" HEIGHT="18" ALIGN="BOTTOM" BORDER="0" SRC="images/img6.gif" ALT="$F$">-processes, increasing processes, base-norm ordered Banach space, Kukutani <IMG WIDTH="19" HEIGHT="18" ALIGN="BOTTOM" BORDER="0" SRC="images/img7.gif" ALT="$L$">-space, Choquet simplex, Bauer simplex, <IMG WIDTH="24" HEIGHT="18" ALIGN="BOTTOM" BORDER="0" SRC="images/img8.gif" ALT="$K$">-simplex, Boolean algebra, Choquet integral representation, Stone correspondence, predictable sets
Article copyright: © Copyright 1983 American Mathematical Society