Corollary 2

— For every integrable set $A$, there exist:

$1^\circ$ a set $A_1 \supset A$, a countable intersection of integrable open sets, such that $A_1 - A$ is negligible;

$2^\circ$ a set $A_2 \subset A$, a countable union of pairwise disjoint compact sets, such that $A - A_2$ is negligible.

$1^\circ$ For every integer $n$, there exists an integrable open set $G_n \supset A$ such that $|\mu|(G_n) - |\mu|(A) \leq 1/n$; if $A_1$ is the intersection of the $G_n$, then $|\mu|(A_1) = |\mu|(A)$ (No. 5, Cor. of Prop. 7), therefore $A_1 - A$ is negligible.

$2^\circ$ Let us define compact sets $K_n$ inductively as follows: $K_1 \subset A$ and $|\mu|(A - K_1) \leq 1$; $K_n \subset A - \bigcup_{i=1}^{n-1} K_i$ and $|\mu|(A \cap \mathbf{C}(\bigcup_{i=1}^{n-1} K_i) \cap \mathbf{C}K_n) \leq 1/n$ for $n > 1$ (Th. 4); if $A_2$ is the union of the $K_n$, then $|\mu|(A_2) = |\mu|(A)$ (No. 5, Prop. 8), therefore $A - A_2$ is negligible.