This is Zygmund lemma 3.15, which proof is left as an exercise:
If $\{I_k\}_{k=1}^N$ is finite collection of nonoverlapping intervals then $\cup I_k$ is measurable and $|\cup I_k| = \sum |I_k|$
I found this proof but I don't get the end of it:
We know that $\overset{N}\bigcup I_k$ is covered by $\overset{M}\bigcup J_n^*$ but that implies $$|\bigcup I_k| \le |\bigcup J_n^*| $$ And I know $ |\bigcup I_k| \le \sum |I_k|$. But how can I get $$\sum |I_k| \le \sum |J_n^*| $$ ?
Thanks in advance.