Remark 2
Tag 08V9, permanent. Cite it as [Bourbaki, Tag 08V9].
Integration, IV, p. 68, § 5. Measurable functions and sets
If the topology of F has a countable base then condition b) of Th. 4 is automatically satisfied for every mapping of X into F. The proof shows, moreover, that it suffices to assume that the inverse images under of the closed balls with rational radii, whose centers belong to a dense countable subset of F, are measurable sets.
Bourbaki puts the proof directly after the statement, so what is above is the statement and its proof together.
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