Corollary 4
Tag 08UW, permanent. Cite it as [Bourbaki, Tag 08UW].
Integration, chapter IV, § 5. Measurable functions and sets
— Let F be a vector space of dimension n over R and let (ek)_{1 \leq k \leq n} be a basis of F. In order that a function f = \sum_{k=1}^{n} e_k f_k be measurable, it is necessary and sufficient that each of the numerical functions f_k be measurable.*
Bourbaki puts the proof directly after the statement, so what is above is the statement and its proof together.
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