Proposition 8

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

Integration, V, p. 47, § 5. Measures defined by numerical densities

Let $\theta$ be a complex measure on $T$ , $g_1$ a locally $\theta$ -integrable complex function, $\theta_1$ the measure $g_1 \cdot \theta$ . For a complex function $g_2$ defined on $T$ to be locally $\theta_1$ -integrable, it is necessary and sufficient that $g_2 g_1$ be locally $\theta$ -integrable, in which case

g_2 \cdot \theta_1 = g_2 \cdot (g_1 \cdot \theta) = (g_2 g_1) \cdot \theta

('associativity formula').

By the corollary of Prop. 4, to say that $g_2$ is $\theta_1$ -measurable is equivalent to saying that $g_2 g_1$ is $\theta$ -measurable. Let us suppose this condition to be satisfied. For every function $f \in \mathscr{K}_+(T)$ we have, by Propositions 2 and 3,

\int^\bullet |g_2| f d|\theta_1| = \int^\bullet |g_2| f |g_1| d|\theta| = \int^\bullet |g_2 g_1| f d|\theta|.

To say that $g_2$ is locally $\theta_1$ -integrable is therefore equivalent to saying that $g_2 g_1$ is locally $\theta$ -integrable. Assuming this condition to be satisfied, by Th. 1 we have

\int f \, d(g_2 \cdot \theta_1) = \int f g_2 \, d\theta_1 = \int f g_2 g_1 \, d\theta = \int f \, d(g_2 g_1 \cdot \theta),

a formula equivalent to (10).

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

Proposition 8

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