Fix $m \in \mathbb{N}$ be such that $\mu(E_m) < \infty$. Result
R1130: Intersection is largest lower bound shows that
\begin{equation}
\bigcap_{n \in \mathbb{N}} E_n
\subseteq E_m
\end{equation}
Since each of $E_0, E_1, E_2, \ldots$ are measurable in $M$, then result
R1030: Sigma-algebra is closed under countable intersections shows that the intersection $\bigcap_{n \in \mathbb{N}} E_n$ is also measurable in $M$. Therefore,
R975: Isotonicity of unsigned basic measure allows us to conclude
\begin{equation}
\mu \left( \bigcap_{n \in \mathbb{N}} E_n \right)
\leq \mu(E_m)
< \infty
\end{equation}
$\square$
