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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 $t\in I$?

tim123
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    Why do you think $A\cap {T \leq t}$ must belong to $\sigma_t$ for every $A$? If a set belongs to a sigma algebra you cannot say that every subset belongs to it. – Kavi Rama Murthy Aug 01 '19 at 10:02
  • ${T\leq t}={w\in \Omega| T(\omega)\leq t}.$ So we know that $A\cap{T\leq t}$ only contains $w \in \Omega$ that forfill the condition. At least this is what i think. – tim123 Aug 01 '19 at 10:18
  • however, ur post makes a lot of sense,while thinking more about it, I think that I understand this $\sigma_T$ much better now. – tim123 Aug 01 '19 at 10:34

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