Theorem 1

Let $\mathbf{f}$ be a function defined on $T$, with values in a Banach space $F$ or in $\overline{\mathbf{R}}$. In order that $\mathbf{f}$ be essentially integrable for the measure $\eta = u \cdot \theta$, it is necessary and sufficient that $uf$ be essentially $\theta$-integrable, in which case

Suppose moreover that $u$ is continuous and that $u(t) \neq 0$ for all $t \in T$; then $f$ is integrable for the measure $\eta$ if and only if $uf$ is integrable for $\theta$.

The case that $u$ and $\theta$ are positive follows at once from Th. 2 of §4, No. 4. The first and the last assertion of the statement then follow immediately, because $f$ is essentially integrable (resp. integrable) with respect to $\eta = u \cdot \theta$ if and only if it is essentially integrable (resp. integrable) for $|\eta| = |u| \cdot |\theta|$. Finally, suppose that $f$ is essentially integrable for $\eta$ (hence for $|\eta|$); we make use of the decomposition (2): $f$ is essentially integrable for each of the measures $\eta_{ij} = g_i \cdot \mu_j$ ($i = 1, 2, 3, 4,\ j = 1, 2, 3, 4$), because these measures are $\leq |\eta|$. We have

The formula (6) follows immediately from this.