If $X:(\Omega\times[0,\infty),\mathcal P)\to(\mathbb R,\mathcal B(\mathbb R))$ is measurable and $f$ is <Borel measurable function>[Borel measurable], then $f\circ X$ is measurable by <composition of measurable functions>. Hence applying a Borel function pointwise to a predictable process again gives a predictable process.

Solution

Past exam of the mathematics course of the University of Cambridge / 2023 / iii / Paper 202 / 2 / d / Solution

 Solution (source code)