Past exam of the mathematics course of the University of Cambridge / 2022 / iii / Paper 202 / 1 / c / Solution

Indicators of the rectanglestogether 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 $L^2(\mathcal{P},\nu )$.