The problem of existence and uniqueness of a Doob-Meyer decomposition for a submartingale which is indexed by a family of subsets of a given locally compact topological space \(T\) is studied. The predictable \(\sigma\)-algebra is introduced and the admissible function (Doléans- Föllmer function) associated with any process is defined. Then some class (D) conditions are given which allow the extension of the admissible function to a \(\sigma\)-additive measure on the predictable \(\sigma\)-algebra. It is proved that such an extension exists (and is unique) for three types of submartingales: submartingales which are squares of martingales; submartingales such that a family of conditional expectations is uniformly integrable; and submartingales such that the family defined by the process stopped at stopping set taking finite number of configurations, is uniformly integrable. Then a Doob-Meyer decomposition is proved: a set-indexed submartingale can be decomposed into the sum of a weak martingale and an increasing process. In assumption about predictability the uniqueness of the decomposition is proved. Finally, some remarks about quasimartingales are discussed.