Predictable sigma-algebra ($\mathcal{P}$)

The predictable sigma-algebra on $\Omega \times [0,\infty )$ is generated by $A\times \{0\}$ for $A\in {\mathcal{F}}_0$ and by $A\times (s,t]$ for $A\in {\mathcal{F}}_s$. Equivalently, it is the smallest sigma-algebra making every left-continuous adapted process measurable.

A simple predictable process is constant on finitely many deterministic time intervals $(t_k,t_{k+1}]$, and its value there is ${\mathcal{F}}_{t_k}$-measurable. Such processes generate the predictable sigma-algebra and are dense in the corresponding $L^p$ spaces for finite measures.

Every deterministic càdlàg function is Borel measurable in time. Since the predictable sigma-algebra contains every set $\Omega \times A$ with $A$ Borel, the corresponding deterministic process is predictable.