= 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 $n\geq1$, define
$$
X_t^{(n)}=X_0\mathbf1_{\{0\}}(t)
+\sum_{k\geq0}X_{k2^{-n}}
\mathbf1_{(k2^{-n},(k+1)2^{-n}]}(t).
$$
On 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.
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