Remark 1

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

Integration, IV, p. 24, § 3. LP spaces

As has already been signalled (§2, No. 5) the $p$ -th power integrable functions with values in $\mathbf{F}$ in general do not form a vector space.

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

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Nothing.