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

Every simple predictable process is left-continuous and adapted: on $(t_k,t_{k+1}]$ its value is already ${\mathcal{F}}_{t_k}$-measurable. Hence any sigma-algebra making all left-continuous adapted processes measurable contains the generators of $\mathcal{P}$.

Conversely, let $X$ be left-continuous and adapted. For $n\ge 1$, defineOn each bounded time interval this is a simple predictable process, and left continuity gives $X_t^{(n)}\to X_t$ for every $t$. Thus $X$ is $\mathcal{P}$-measurable. This proves that $\mathcal{P}$ is exactly the smallest sigma-algebra making every left-continuous adapted process measurable.