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

If $f:(X,\mathcal A)\to(Y,\mathcal B)$ and $g:(Y,\mathcal B)\to(Z,\mathcal C)$ are measurable, then $g\circ f$ is measurable because
$$
(g\circ f)^{-1}(C)=f^{-1}(g^{-1}(C)).
$$


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 Incoming links: Composition of measurable functions