Hi i have following defnition
$T:\Omega\rightarrow I$
$\sigma_T=\{A\in\Omega|A \cap\{{T \leq t\}}\in\sigma_t\, \text{for all t}\in I \}$
I do not understand the sense of the definition. Why there has to be the condition
$A\in\Omega|A \cap\{{T \leq t\}}$ since $A \cap\{{T \leq t\}}$ should be $\in \sigma_t$ for all $A\in \Omega$, because we know that $\{T\leq t\}\in \sigma_t$ ?
Further I am confused of "for all $t\in I$." Does it mean that I have to consider $\bigcap_{t\in I}A\cap\{T\leq t\}\in \sigma_t$ since we want it to have it for all $t\in I$?