= Solution
Fix $t>0$ and partition $[0,t]$ into $m$ equal intervals, with $r_k=kt/m$. Round $t\wedge T$ upward within this interval. The corresponding sampled value is
$$
Y_m=X_0\mathbf1_{\{T=0\}}+\sum_{k=1}^{m-1}X_{r_k}\mathbf1_{\{r_{k-1}<T\leq r_k\}}+X_t\mathbf1_{\{T>r_{m-1}\}}.
$$
Every time in this sum is at most $t$. The <stopping time> property makes each <indicator function> $\mathcal F_t$-<measurable>, while the <adapted process> property makes each sampled value $\mathcal F_t$-<measurable>. Consequently $Y_m$ is $\mathcal F_t$-<measurable>.
The rounded times approach $t\wedge T$ from the right, so right continuity gives $Y_m\to X_{t\wedge T}$ for the chosen pathwise <càdlàg> version. If path regularity is instead stated only <almost surely>, the usual complete <filtration> handles the exceptional <null set>; on an incomplete <filtration> one should formulate that case as existence of an <adapted> version. At $t=0$, the value is simply $X_0$. Thus \b[$X^T$ is an <adapted process>], by <adaptedness of a stopped right-continuous process>. The argument actually needs neither the <martingale> property nor boundedness of $T$.
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