Proposition 10

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

— For an open (resp. closed) set $A$ in $X$ to be integrable, it is necessary and sufficient that $|\mu|^*(A) < +\infty$ .

Since $\varphi_A$ is then lower (resp. upper) semi-continuous, the proposition follows from Prop. 5 of No. 4 and its Corollary 1.

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

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