Definition

We denote by $\mathcal{E}$ the set of all processes of the form $$\tag{★}X=\sum_{i=1}^{n}a_{i}\mathbb{1}_{(s_{i},t_{i}]\times F_{i}}+\sum_{k=1}^{n}d_{k}\mathbb{1}_{\{ 0 \}\times F_{0k}}$$where $0\le s_{i}<t_{i}, F_{i}\in\mathcal{F}_{s_{i}}, \forall i=1,\dots,n$ and $F_{0k}\in\mathcal{F}_{0}, \forall k=1,\dots,n$. We call $\mathcal{E}$ the set of $\mathcal{R}$-simple processes or the space of elementary predictable processes

Note

In (★) all the $(s_{i},t_{i}]\times F_{{i}}$ can be assumed WLOG to be pairwise disjoint.