This is part of the solution to problem 3.2 (i) in Karatzas and Shreve. Here, $T_1, T_2, \dots$ is a sequence of independent exponentially distributed random variables with parameter $\lambda>0$. $S_0 = 0, S_n = \sum_{i=1}^n T_i$. And $N_t = \max\{n \ge 0: S_n \le t\}$ for $0\le t < \infty$.
I have question about the last sentence of the following excerpt. I know that $\mathscr{H}$ is generated by sets of the form $\{T_1 \le t_1, \dots , T_n \le t_n, N_s = n\}$ since the $\sigma$-field $\sigma(T_1, \dots, T_n)$ are generated by such sets $\{T_1 \le t_1, \dots , T_n \le t_n\}$. However, how is it generated by $\{S_1 \le t_1, \dots, S_{n-1}\le t_{n-1}\}$?
