= Solution
A <simple predictable process> has the form
$$
H(\omega,t)=H_0(\omega)\mathbf1_{\{0\}}(t)
+\sum_{k=0}^{m-1}H_k(\omega)\mathbf1_{(t_k,t_{k+1}]}(t),
$$
where $0=t_0<\cdots<t_m<\infty$, $H_0$ 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.
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