Let $(X\_t)$ a progressively measurable process, i.e. $[0,t]\times \Omega \ni (t,\omega )\mapsto X\_t(\omega )\in (\mathbb R, \mathcal B(\mathbb R))$ is $\mathcal B([0,t])\otimes \mathcal F$ measurable. Prove that $X\_T$ is $\mathcal F\_T$ measurable on $\{T<\infty \}$.

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**Q1)** What does it mean that $X\_T$ is $\mathcal F\_T$ measurable on $\{T<\infty \}$ ? Does it mean that for all Borel set $B$, we have $(X\_T\boldsymbol 1\_{\{T<\infty \}})^{-1}(B)\in \mathcal F\_T$ ?

Suppose my definition is correct. Then that's enough to prove 

$\{X\_T\boldsymbol 1\_{\{T<\infty \}}\leq x\}\in \mathcal F\_T$ for all $x$. **($\*$)**

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**Q2)** Any idea on how to prove **($\*$)** ? 

I don't see how I can prove that 

$\{X\_T\boldsymbol 1\_{\{T<\infty \}} \} \in \mathcal F\_\infty $ and $\{X\_T\boldsymbol 1\_{\{T<\infty \}}\leq x\}\cap \{T\leq t\}\in \mathcal F\_t$ for all $t$.