Since $F_0, F_1, F_2, \dots$ partitions $X$, then applying disjoint additivity and result
R237: Intersection distributes over union, one has
\begin{equation}
\begin{split}
\mu (E) = \mu (E \cap X)
& = \mu \left( E \cap \bigcup_{n \in \mathbb{N}} F_n \right) \\
& = \mu \left( \bigcup_{n \in \mathbb{N}} (E \cap F_n) \right)
= \sum_{n \in \mathbb{N}} \mu (E \cap F_n)
\end{split}
\end{equation}
$\square$
