= Solution
Observing $X_s$ reveals exactly whether $\omega<s$. Thus the <natural filtration> is
$$
\mathcal F_t=\sigma\{(0,s):0\leq s\leq t\}.
$$
Every generating <set> either contains the entire tail $[t,\infty)$ or avoids it, so this tail remains one indistinguishable atom. Conversely, these initial intervals generate the full relative <Borel sigma-algebra> on $(0,t)$: use rational endpoints below $t$, together with $(0,t)$ itself. Therefore every <Borel set> contained in $(0,t)$, and its complement, belongs to $\mathcal F_t$. We obtain
$$
\boxed{\mathcal F_t=\{B\in\mathcal F:B\subseteq(0,t)\ \text{or}\ B^c\subseteq(0,t)\}.}
$$
The strict endpoint is essential: the value at $\omega=t$ is still $X_t=1$, so it cannot be distinguished from a later lifetime.
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