Definition ($\mathcal{E}$, simple predictable processes)

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

Definition ($\Lambda$, the set of predictable locally integrable processes)

Let $M$ be a right continuous $L^{2}$-martingale. We denote by $$\Lambda^{2}(\mathscr{P},M)$$the set of all RVs $X:(\mathbb{R}^{+}\times\Omega,\mathscr{P})\to(\mathbb{R},\mathcal{B}(\mathbb{R}))$ that are measurable (i.e. $X$ is a predictable process) s.t. $$\mathbb{1}_{[0,t]}X\in L^{2}(\mathbb{R}^{+}\times\Omega,\mathscr{P},\mu_{M}),\quad t\ge 0$$