In this paper we present the concept of a weak Banach-valued quasi-martingale. The concept of a strong Banach-valued quasi-martingale was introduced by the author in an earlier paper. We prove that a weak Banach-valued quasi-martingale under certain conditions has a weak Riesz decomposition, i.e., it can be decomposed in an essentially unique way as a sum of a weak martingale and a weak quasi-potential.
"Riesz Decomposition for Weak Banach-Valued Quasi-Martingales." Ann. Math. Statist. 43 (3) 1020 - 1026, June, 1972. https://doi.org/10.1214/aoms/1177692568