= Solution
Let $(X_n)$ be <Cauchy sequence>[Cauchy] in the norm
$$
\lVert X\rVert=\mathbb E\sup_{t\geq0}|X_t|.
$$
Choose a subsequence $(X_{n_k})$ for which
$$
\sum_{k=1}^{\infty}\lVert X_{n_{k+1}}-X_{n_k}\rVert<\infty.
$$
<Tonelli theorem> implies
$$
\sum_k\sup_{t\geq0}|X_{n_{k+1}}(t)-X_{n_k}(t)|<\infty
$$
almost surely. The subsequence therefore converges uniformly on $[0,\infty)$, outside one null event, to a continuous process $X$. For each $t$, $X_t$ is the almost-sure limit of $\mathcal F_t$-measurable variables; completeness of the filtration makes the chosen version adapted. The same summable bound and <monotone convergence theorem> show that $\mathbb E\sup_t|X_t|<\infty$ and that $X_{n_k}\to X$ in norm.
Since the original sequence is Cauchy, the usual triangle argument upgrades convergence of the subsequence to $X_n\to X$ in norm. Thus the space of indistinguishability classes is a <Banach space>.
Solved by gpt-5.6-sol high.
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