I am given the following definition:
T is a stopping time if $\{T⩽t\}∈F_t$ for all t. $\{S∧T⩽t\}=\{S⩽t\}∪\{T⩽t\}$ both of which are in $F_t$.
I do not understand the $\cup$ sign.
In my logic (which seems wrong), I would follow that the $\cup$ should be a $\cap$ sign. I want that both T and S are $\leq$ t (that is what the $\land$ sign means to me), so I would assume I have to intersect the respective sets (Would my logic at least be right for 'standard' logical sets?).
I guess it has something to do with the filtration $F_t$ but I'm at a total loss here, please help.
The equality rests on the fact that for $u,v\in\mathbb R$, $$ \min\{u,v\}\leqslant t\Leftrightarrow \left(\left(u\leqslant t\right)\mbox{ or }\left(v\leqslant t\right)\right). $$ Similarly, $$ \max\{u,v\}\leqslant t\Leftrightarrow \left(\left(u\leqslant t\right)\mbox{ and }\left(v\leqslant t\right)\right). $$