 Solution

Observing $X_s$ reveals exactly whether $\omega <s$. Thus the natural filtration isEvery 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 obtainThe strict endpoint is essential: the value at $\omega =t$ is still $X_t=1$, so it cannot be distinguished from a later lifetime.