I've seen a proof skatch of the discussed statement below, but I don't really understand it. Does anyone know what is the elaborated proof, or where can I find it?
$X$ is a progressive measurable function, where $\mathcal{F}$ is the natural filtration of $X$. $\tau$ is an $\mathcal{F}$-stopping time.
Probably the following is not the best translation, but we say $\left(\Omega,\mathcal{A},\mathbf{P},\mathcal{F}\right)$ is complete corresponding to stopping, if for any $\sigma$ $\mathcal{F}$-stopping time and for all $\omega$ there exist an $\alpha\left(\omega\right)=\omega'$ where
$$X^{\sigma}\left(t,\omega\right)=X\left(t,\alpha\left(\omega\right)\right),$$
where $X^{\sigma}$ is the stopped process at $\sigma$.
We say $\omega\sim_{\tau}\omega'$ if the corresponding trajectories are the same on $\left[0,\tau\right]$, i.e. $X^{\tau}\left(t,\omega\right)=X^{\tau}\left(t,\omega'\right)$.
Let's define three $\sigma$-algebras:
1, $\mathcal{F_{\tau}}$: $A\in\mathcal{A}$ is said to be in $\mathcal{F}_{\tau}$ if for all $t$
$$A\cap{(\tau\leq t)} \in\mathcal{F}_{t}.$$
2, $\mathcal{H}_\tau$: $A\in\mathcal{F}_{\infty}$ is said to be in $\mathcal{H}_{\tau}$ if and only if $A=\left[A\right]_{\tau}$, where $\left[A\right]_{\tau}$ denotes the set of those $\omega'$ outcomes, where there exist $\omega\in A$, such as $\omega\sim_{\tau}\omega'$.
3, $\mathcal{G}_{\tau}$: This sigma-algebra is defined as
$$\mathcal{G}_{\tau}\overset{\cdot}{=}\sigma\left(X\left(\tau\land t:\forall t\right)\right).$$
The statement is the following: if $\Omega$ is complete corresponding to stopping, then
$$\mathcal{F}_{\tau}=\mathcal{H}_{\tau}=\mathcal{G}_{\tau}.$$
The proof goes like to prove the following:
$$\mathcal{F}_{\tau}\subseteq\mathcal{H}_{\tau}\subseteq\mathcal{G}_{\tau}\subseteq\mathcal{F}_{\tau}.$$
I have problem with $\mathcal{F}_{\tau}\subseteq\mathcal{H}_{\tau}$, I think I understand the proof of $\mathcal{H}_{\tau}\subseteq\mathcal{G}_{\tau}\subseteq\mathcal{F}_{\tau}$.