Result
R292: Union is smallest upper bound shows that
\begin{equation}
E_1, \ldots, E_N \subseteq \bigcup_{n = 1}^N E_n
\end{equation}
Thus, if $\mu(\bigcup_{n \in \mathbb{N}} E_n) < \infty$, then result
R975: Isotonicity of unsigned basic measure allows us to deduce
\begin{equation}
\mu(E_1), \ldots, \mu(E_n)
\leq \mu \left( \bigcup_{n = 1}^N E_n \right)
< \infty
\end{equation}
Conversely, if $\mu(E_1), \ldots, \mu(E_N) < \infty$, then result
R979: Countable subadditivity of measure gives the upper bound
\begin{equation}
\mu \left( \bigcup_{n = 1}^N E_n \right)
\leq \sum_{n = 1}^N \mu(E_n)
< \infty
\end{equation}
$\square$
