Corollary 2

Tag 08S6, permanent. Cite it as [Bourbaki, Tag 08S6].

Integration, chapter IV, § 4. Integrable functions and sets

— If the $a_k$ ( $1 \leq k \leq n$ ) are vectors in $F$ and the $f_k$ ( $1 \leq k \leq n$ ) are integrable numerical functions, then the function $f = \sum_{k=1}^n a_k f_k$ is integrable and

\int \left( \sum_{k=1}^n a_k f_k \right) d\mu = \sum_{k=1}^n a_k \int f_k \, d\mu.

Bourbaki puts the proof directly after the statement, so what is above is the statement and its proof together.

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