Show that Theorem 5.7 is still valid, if (left(G_{n}ight)_{n in mathbb{N}} subset mathscr{G}) is not an increasing

Question:

Show that Theorem 5.7 is still valid, if $\left(G_{n}ight)_{n \in \mathbb{N}} \subset \mathscr{G}$ is not an increasing sequence but any countable family of sets such that

\[\text { (1) } \bigcup_{n \in \mathbb{N}} G_{n}=X \quad \text { and } \quad \text { (2) } \quad u\left(G_{n}ight)=\mu\left(G_{n}ight)

[set $F_{N}:=G_{1} \cup \cdots \cup G_{N}=F_{N-1} \cup G_{N}$ and check by induction that $\mu\left(F_{N}ight)=u\left(F_{N}ight)$; use then Theorem 5.7 .]

Data from theorem 5.7