Let $M = (X, \mathcal{F}, \mu)$ be a D1158: Measure space.
Let $E_0, E_1, E_2, \ldots \in \mathcal{F}$ each be a D1109: Measurable set in $M$ such that

(i)	\begin{equation} \exists \, n \in \mathbb{N} : \mu(E_n) < \infty \end{equation}

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} $\square$