Let \((\Omega, {\mathcal A}, P)\) be a complete probability space and let \(({\mathcal A}_ n)\) be an increasing sequence of \(sub-\sigma\)-algebras of the \(\sigma\)-algebra \({\mathcal A}\). A sequence \((X_ n)\) is called an \(s\)- game if for each \(\varepsilon>0\) \(\exists p\) such that \(\forall q\in\mathbb{N}\), \(s\in S\) with \(s\geq q\geq p\), \(P(\| X_ q(s)- X_ q\| >\varepsilon) <\varepsilon\) holds. A sequence \((X_ n)\) is called an \(L^ 1\) \(s\)-game if it is an \(S\)-game and the subsequence \((X_ s)\) is an \(L^ 1\)-amart. Here, \((X_ n)\) is a sequence of \(E\)-valued Bochner integrable random variables for a fixed Banach space \(E\).
The following results are proved:
(i) A sequence \((X_ n)\) is an \(L^ 1\) \(s\text{-game} \Leftrightarrow (X_ n)\) has a Riesz decomposition \(X_ n= M_ n+ P_ n\), where \((M_ n)\) is a martingale, \((P_ n)\) converges to zero in probability and the subsequence \((P_ s)\) is an \(L^ 1\)-potential.
(ii) The class of \(L^ 1\)-bounded \(L^ 1\) \(s\)-games in \(\ell^ 1\) is a Banach lattice with the norm \(\|(X)\|_ \mathbb{N} =\sup_{(\mathbb{N})} E(\| X_ n \|)\), where \(\mathbb{N}\) is the set of all positive integers.