Q: Royden Chapter 2, Theorem 11

Question

I don't understand the first part. Why can the set $E$ (subset of $\mathbb{R}$) be expressed as the disjoint union of a countable collection of measurable sets, each of which has finite outer measure? I searched for a proof of this statement in the book but I didn't found any. Can someone explain to me why is this true?

Answer 1

For each integer $n$, define $I_n := (n, n+1]$. These sets are disjoint, each have finite measure, and we have $\mathbb{R} = \cup_n I_n$. Now let $J_n := I_n \cap E$. These sets have finite measure and are still pairwise disjoint, and we have $E = \cup_n J_n$.

Answer 2

Royden proves this earlier on Page 37.