Corollary 3
Tag 08T1, permanent. Cite it as [Bourbaki, Tag 08T1].
Integration, chapter IV, § 4. Integrable functions and sets
— Every set of finite outer measure is contained in the union of a negligible set and a countable family of pairwise disjoint compact sets the sum of whose measures is finite.
It suffices to apply Corollary 2 to an integrable open set containing the given set.
Bourbaki puts the proof directly after the statement, so what is above is the statement and its proof together.