Let $T>0$ and $\{\mathcal{F_s}\}_{s\in[0,T]}$ be a filtration on the space $\Omega$. Define,$$T' = \sum_{n=0}^k\,t_n\,1_{X_n}\,,$$ where $0=t_0<t_1<...<t_k=T$ and $\{X_n\}$ be a partition of $\Omega$ and $X_n\in \mathcal{F_{t_n}}$ for all $n$.
I want to prove that $T'$ is $\{\mathcal{F_s}\}$ stopping time. i.e: $\{T'=s\}\in \mathcal{F_s} $ for all $s\in[0,T]$.
Let $s$ be fixed and define $T_n = t_n1_{X_n}$ for each $n$. I think it suffices to prove that $T_n$ is $\mathcal{F_s}$ m'ble but not sure how to proceed. Any hint on how to proceed.
Your definition of stopping time applies to discrete case but here you have the continuous case. The definition is $\{T'\leq s\} \in \mathcal F_s$ for all $s$. Verify that $\{T'\leq s\}=\bigcup_I X_n$ where $I=\{n: t_n \leq s\}$. Note that$X_n \in \mathcal F_{t_n} \subseteq \mathcal F_s$ for any $n$ with $t_n \leq s$. Hence $\{T'\leq s\} \in \mathcal F_s$.