Corollary 2

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

Integration, chapter IV, § 3. LP spaces

— For a positive numerical function $f$ to belong to $\mathscr{L}^p$ , it is necessary and sufficient that $f^p$ belong to $\mathscr{L}^1$ .

Note that if $f$ is a numerical function of arbitrary sign, such that $|f|^p$ belongs to $\mathscr{L}^1$ , $f$ does not necessarily belong to $\mathscr{L}^p$ (cf. § 4, Exer. 8).

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

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Cites

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