Remark 2

Tag 09QY, permanent. Cite it as [Bourbaki, Tag 09QY].

Integration, IX, p. 58, § 5. Measures on completely regular spaces

If the measure $\mu$ is bounded, the formula $\mu^*(f) = \inf_{g \in S_f} \mu(g)$ is obviously valid for every function $f$ that is upper semi-continuous, positive and bounded.

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

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