Definition 2

Tag 088L, permanent. Cite it as [Bourbaki, Tag 088L].

Functions of a Real Variable, chapter II, § 2. INTEGRALS OVER NON-COMPACT INTERVALS

One says that the integral of a regulated function $\mathbf{f}$ over an interval $I \subset \mathbf{R}$ is absolutely convergent if the integral of the positive function $\| \mathbf{f}(x) \|$ is convergent.

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

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