Abstract
Set-indexed martingales and submartingales are defined and studied. The admissible function of a submartingale is defined and some class (D) conditions are given which allow the extension of the function to a σ-additive measure on the predictable σ-algebra. Then, we prove a Doob-Meyer decomposition: A set-indexed submartingale can be decomposed into the sum of a weak martingale and an increasing process. A hypothesis of predictability ensures the uniqueness of this decomposition. An explicit construction of the increasing process associated with a submartingale is given. Finally, some remarks, about quasimartingales are discussed.
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