Let $P = (\Omega, \mathcal{F}, \mathbb{P}, T)$ be a D2839: Probability-preserving system such that

(i)	$E \in \mathcal{F}$ is an D1716: Event in $P$
(ii)	\begin{equation} \mathbb{P} \left( E \triangle T^{-1} E \right) = 0 \end{equation}

Let $n \in \mathbb{N}$ be a D996: Natural number.

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

(i)	R2090: Isotonicity of probability measure
(ii)	R4456:

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. $\square$