Let $S,T$ be continuous stopping time with respect to filtration $\mathcal{G}_t$. If $\Lambda\in \mathcal{G}_{S\lor T}$, then $\Omega\cap \{S\leq T\}\in \mathcal{G}_T$
This problem is from Diffusion Markov Process and Martingale. Here is what I try...
$\Lambda\in \mathcal{G}_{S\lor T}$ iff $\Lambda\cap \{S\leq t\} \cap \{T\leq t\}\in \mathcal{G}_t$.
$\Lambda\cap \{S\leq T\}\in \mathcal{G}_{ T}$ iff $\Lambda\cap \{S\leq T\} \cap \{T\leq t\}\in \mathcal{G}_t$
So I try to write $S\leq T$ as $\{S\leq r\}\cap \{r\leq T\}$ where $r\in \mathbb{R}$. But from here I don't know how to preceed.
Does anyone have any idea?
Thankes in advance
I think I have figured out myself..
Let $A\in \mathcal{F}_{S\lor T}$. we have \begin{align*} A\cap \{S\leq T\}\cap \{T\leq t\} = [A\cap \{S\leq t\}\cap \{T\leq t\}]\cap [\{S\leq T\}\cap \{T\leq t\}] \end{align*} The first part is in $\mathcal{F}_T$. We know that \begin{align*} \{S>T\}\cap \{T\leq t\}=\cup_{q\in \mathbb{Q}^+\cap [0,t]}(\{S>q\}\cap \{T\leq q\}\cap \{T\leq t\}) \end{align*} The RHS is in $\mathcal{F}_T$. Since $(T\leq t)\in \mathcal{F}_t$, we get $\{S\leq T\}\cap \{T\leq t\}\in \mathcal{F}_t$.