Remark 5
Tag 09PW, permanent. Cite it as [Bourbaki, Tag 09PW].
Integration, IX, p. 45, § 3. Measures and additive set functions
One can show that if I is additive, that is, satisfies (4) for finite sequences, and if I is inner regular, then I is countably additive (Exer. 7). The reader can also ascertain that only additivity and inner regularity are used in the proof of Th. 2 below.
Bourbaki puts the proof directly after the statement, so what is above is the statement and its proof together.
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Nothing.