Corollary

Tag 092I, permanent. Cite it as [Bourbaki, Tag 092I].

Let $g_1$ and $g_2$ be two locally $\mu$ -integrable numerical functions; then

\inf(g_1 \cdot \mu, g_2 \cdot \mu) = \inf(g_1, g_2) \cdot \mu ; \quad \sup(g_1 \cdot \mu, g_2 \cdot \mu) = \sup(g_1, g_2) \cdot \mu .

In particular, if $g$ is a locally $\mu$ -integrable numerical function, then

(g \cdot \mu)^+ = g^+ \cdot \mu ; \quad (g \cdot \mu)^- = g^- \cdot \mu .

This follows at once from Prop. 2 and the formulas (6) of Ch. II, §1, No. 1.

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

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