Density of simple predictable processes for finite measures (${\overline{\mathcal{S}}}^{L^2(\mu )}=L^2(\mathcal{P},\mu )$)

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.