Remark 3
Tag 09FE, permanent. Cite it as [Bourbaki, Tag 09FE].
Integration, VII, p. 36, § 2. Quotient of a space by a group; homogeneous spaces
Cor. 3 c) of Prop. 6 shows at once that the results of this subsection remain valid in the case of complex measures (except for those that involve the upper integral).
Bourbaki puts the proof directly after the statement, so what is above is the statement and its proof together.
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Nothing.