Lemma 1.19 in Jacod and Shiryaev's Limit Theorems for Stochastic Processes states the following:
Any stopping time $T$ on the completed stochastic basis $(\Omega,\mathcal{F}^P,\mathbf{F}^P,P)$ is a.s. equal to a stopping time on $(\Omega,\mathcal{F},\mathbf{F},P)$.
Here $\mathbf{F}=(\mathcal{F}_t)_{t\geq 0}$ is a filtration and $(\Omega,\mathcal{F}^P,\mathbf{F}^P,P)$ denotes the $P$-completion of $(\Omega,\mathcal{F},\mathbf{F},P)$. They construct a stopping $T'$ with respect to $(\Omega,\mathcal{F},\mathbf{F},P)$ which satisfies $ \{T