Using the result
R4452: Symmetric difference of countable unions is subset of union of symmetric differences, we have the inclusion
\begin{equation}
E \triangle \bigcup_{m = n}^{\infty} T^{-m} E
\subseteq \bigcup_{m = n}^{\infty} (E \triangle T^{-m} E)
\end{equation}
Applying this as well as the results
we find that
\begin{equation}
0
\leq \mathbb{P} \left( E \triangle \bigcup_{m = n}^{\infty} T^{-m} E \right)
\leq \mathbb{P} \left( \bigcup_{m = n}^{\infty} (E \triangle T^{-m} E) \right)
\leq \sum_{m = n}^{\infty} \mathbb{P}(E \triangle T^{-m} E)
= 0
\end{equation}
That is, $\mathbb{P} ( E \triangle \bigcup_{m = n}^{\infty} T^{-m} E ) = 0$. The claim now follows as a consequence of
R4453: Probability of symmetric difference of event and subevent. $\square$
