 Solution

A simple predictable process has the formwhere $0=t_0<\cdots <t_m<\infty$, $H_0$ is bounded and ${\mathcal{F}}_0$-measurable, and each $H_k$ is bounded and ${\mathcal{F}}_{t_k}$-measurable. Finite linear combinations of indicators of predictable rectangles are of this form after refining the finitely many time partitions.

Those rectangles form a semiring of sets generating $\mathcal{P}$. The collection of sets whose indicators can be approximated in $L^2(\nu )$ by simple predictable processes is a monotone class: for an increasing sequence, truncate the union and use the finiteness of $\nu$; complements and finite disjoint unions are handled by linearity. The Monotone class theorem therefore puts every $\mathcal{P}$-measurable indicator in the closure. Ordinary measurable simple functions are dense in $L^2(\mathcal{P},\nu )$, so simple predictable processes are dense there as well.