Filtration of stopping time equal to the natural filtration of the stopped process
The "saturation" property assumed by Shiryaev is indeed clear for $\Omega$ consisting of the space of cadlag paths from $[0,\infty)$ to $\Bbb R^d$, and $X_t(\omega)=\omega(t)$ for $t\ge0$ and $\omega\in\Omega$. For, given $t\ge 0$ and $\omega\in\Omega$, define $\omega'$ to be the path "$\omega$ stopped at time $t$"; that is, $\omega'(s):=\omega(s\wedge t)$ for $s\ge 0$. This path $\omega'$ is clearly an element of $\Omega$, and $X_{s\wedge t}(\omega) =\omega(s\wedge t) =\omega'(s)=X_s(\omega')$ for each $s\ge 0$.