 Solution

The whole space belongs to ${\mathcal{F}}_T$, since $\Omega \cap \{T\le n\}=\{T\le n\}\in {\mathcal{F}}_n$. If $A\in {\mathcal{F}}_T$, thenIf $A_j\in {\mathcal{F}}_T$, then $({\bigcup }_j{A}_j)\cap \{T\le n\}={\bigcup }_j(A_j\cap \{T\le n\})$ is also in ${\mathcal{F}}_n$. These are precisely the sigma-algebra closure properties, so ${\mathcal{F}}_T$ is a sigma-algebra.

For every integer $m\ge 0$,Hence $\{T\le m\}\in {\mathcal{F}}_T$. Such threshold sets generate the Borel sigma-algebra on ${\mathbb{N}}_0\cup \{\infty \}$, proving $T$ is ${\mathcal{F}}_T$-measurable. This is the stopping-time sigma-algebra.