Proposition 2
Tag 0968, permanent. Cite it as [Bourbaki, Tag 0968].
Integration, chapter V, § 7. Integration with respect to an induced measure
— For a mapping g of X into a topological space to be *-measurable, it is necessary and sufficient that* g be *-measurable in* X *(§5, No. 3, Example)*.
This follows from Prop. 3 of §4.
Bourbaki puts the proof directly after the statement, so what is above is the statement and its proof together.