Assume $X=\left\{ X_{t},\mathscr{F}_{t}^{X}:0\le t<\infty\right\}$ is a strong Markov process with initial distribution $\mu$ on $\left(\Omega,\mathscr{F}_{\infty}^{X},P^{\mu}\right)$. And define $$\mathscr{N}_{t}^{\mu}=\left\{ F\subset\Omega:\exists G\in\mathscr{F}_{\infty}^{X},F\subset G,P^{\mu}\left(G\right)=0\right\} $$Then define the augmentation of $\left\{ \mathscr{F}_{t}^{X}\right\} $ as $\mathscr{F}_{t}^{\mu}=\sigma\left(\mathscr{F}_{t}^{X}\cup\mathscr{N}^{\mu}\right)$. Then $\left\{ \mathscr{F}_{t}^{\mu}\right\}$ is right-continuous. (Proposition 2.7.7 in Karatzas/Shreve in the book "Brownian Motion and Stochastic Calculus")
Now I want to show that any optional time $S$ of $\left\{ \mathscr{F}_{t}^{\mu}\right\}$ is also a stopping time of this filtration, and for each such $S$ there exists an optional time $T$ of $\left\{ \mathscr{F}_{t}^{X}\right\} $ with $\left\{ S\ne T\right\} \in\mathscr{N}^{\mu}$.
As $\left\{ \mathscr{F}_{t}^{\mu}\right\}$ is right-continuous, any optional time $S$ of $\left\{ \mathscr{F}_{t}^{\mu}\right\}$ is also a stopping time of this filtration. But how do I show the second part?