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

If $X:(\Omega \times [0,\infty ),\mathcal{P})\to (\mathbb{R},\mathcal{B}(\mathbb{R}))$ is measurable and $f$ is 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.