Fix $N \in \mathbb{N}$. Result R1130: Intersection is largest lower bound yields the inclusion \begin{equation} \bigcap_{n = 0}^{\infty} \bigcup_{m = n}^{\infty} E_m \subseteq \bigcup_{m = N}^{\infty} E_m \end{equation} Applying now the results

(i)	R975: Isotonicity of unsigned basic measure
(ii)	R979: Countable subadditivity of measure
(iii)	R2933: Isotonicity of countably infinite real summation

we have the inequalities \begin{equation} \mu \left( \bigcap_{n = 0}^{\infty} \bigcup_{m = n}^{\infty} E_m \right) \leq \mu \left( \bigcup_{m = N}^{\infty} E_m \right) \leq \sum_{m = N}^{\infty} \mu(E_m) \leq \sum_{m = 0}^{\infty} \mu(E_m) < \infty \end{equation} Since $\sum_{m = 0}^{\infty} \mu(E_m) < \infty$, then we can apply result R4488: Tails of convergent unsigned basic real series converge to zero to establish the limit \begin{equation} \lim_{N \to \infty} \sum_{m = N}^{\infty} \mu(E_m) = 0 \end{equation} Since $\mu \geq 0$ then this, the above bound, and result R1096: Squeeze theorem for basic sequences imply \begin{equation} \mu \left( \bigcap_{n = 0}^{\infty} \bigcup_{m = n}^{\infty} E_m \right) = 0 \end{equation} Finally, we can use R2371: Equivalent characterisations of membership in a limit superior for sequences of sets to connect the two claims as follows \begin{equation} \mu \left( \bigcap_{n = 1}^{\infty} \bigcup_{m = n}^{\infty} E_m \right) = \mu(\{ x \in X : \# \{ n \in \mathbb{N} : x \in E_n \} = \infty \}) = 0 \end{equation} $\square$