Solution
= Solution
Indicators of the rectangles
$$
A\times(s,t],\qquad A\in\mathcal F_s,
$$
together with $A\times\{0\}$ generate the <predictable sigma-algebra>. Their finite linear span is precisely the set of simple processes. The monotone-class theorem therefore makes this span dense among bounded predictable functions in measure. Since $\nu$ is finite, truncation followed by bounded approximation proves density in $L^2(\mathcal P,\nu)$.