 Solution

A deterministic càdlàg function is Borel measurable. The sets $\Omega \times (s,t]$ and $\Omega \times \{0\}$ show that $\mathcal{B}([0,\infty ))$ is contained in $\mathcal{P}$. Therefore every deterministic càdlàg function, regarded as a process constant in $\omega$, is predictable.