Definition (Predictable)

Let $X=(X_{n})_{n\in\mathbb{N}}$ be a process. Let $(\mathcal{F}_{n})_{n\in\mathbb{N}}$ be a filtration on $(\Omega,\mathcal{F},P)$. $X$ is a $(\mathcal{F}_{n})_{n\in\mathbb{N}}$-predictable process if $$\begin{align*} X_{0}\text{ is }\mathcal{F}_{0}-\text{measurable and }X_{n}\text{ is }\mathcal{F}_{n-1}-\text{measurable}, \ \forall n&\ge 1 \end{align*}$$

Definition ($\mathcal{R}$, the set of predictable rectangles)

We denote $\mathcal{R}$ as the set of subsets of $\mathbb{R}^{+}\times\Omega$ of the form $\{ 0 \}\times F_{0}$ with $F_{0}\in\mathcal{F}_{0}$ and $(s,t]\times F,0\le s<t, \forall F\in\mathcal{F}_{s}$. $\mathcal{R}$ is called the set of predictable rectangles.

Definition ($\mathcal{A}$, the ring generated by $\mathcal{R}$)

Recall the Predictable sets, $\mathcal{R}$. We denote by $\mathcal{A}$ the ring generated by $\mathcal{R}$, i.e. the closure under finite union and proper complement.

Definition ($\mathscr{P}$, predictable $\sigma$-algebra)

We denote by $\mathscr{P}$ the σ-algebra on $\mathbb{R}^{+}\times\Omega$ generated by the predictable rectangles, $\mathcal{R}$ i.e. generated by sets of the form$$\begin{align*} &\{ 0 \}\times F_{0},&F_{0}\in\mathcal{F}_{0}\\ &(s,t]\times F, &\forall0\le s<t,F\in\mathcal{F} \end{align*}$$