Thanks! Then yes, I guess I will give a short proof rather than a reference.

Thanks for the answer. Do you have any reference for this approach?

Moreover I don't see how the series (107) helps in solving the question. The random variable $W$ does not appear in (*), so we might choose any random variable like that which is independent of the process, and all the mutual information would always be $0$, making the series (107) summable.

Thank you @Yuval for your answer. However I am still confused and cannot see how to easily pass from the proof in [2] to the conclusion. Could you please detail a bit more? From my understanding, in [2] they control the reminder $E[\sup_{t\in T}(X_t-X_{\pi_n(t)})]$ by using that it is $0$ if $T$ is finite and $n$ large enough. But to use this argument we would need something like $\lim_{n\to\infty}\sup_k E[\sup_{t\in T^{(k)}}(X_t-X_{\pi_n(t)})]=\sup_k\lim_{n\to\infty} E[\sup_{t\in T^{(k)}}(X_t-X_{\pi_n(t)})]=0$, where $T^{(k)}$ are finite subset whose union is $T_0$. But how to show it?

Thank you, this looks like the reference I need!