Simple predictable process

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.

For every finite measure $\mu$ on the predictable sigma-algebra, bounded simple predictable processes are dense in $L^2(\mu )$. Include a bounded ${\mathcal{F}}_0$-measurable value supported at time zero when $\mu$ may charge that slice. Indicators of the generating rectangles and the Pi-lambda theorem prove density. This gives the completion step in the Itô isometry construction.