Proposition 9

Suppose $g \geq 0$ and locally $\mu$-integrable. Then, the image under $G$ of the positive measure $g \cdot \mu$ is Lebesgue measure on $G(I)$. For a function $f$, defined on $G(I)$, with values in $\overline{\mathbf{R}}$ or in a Banach space, to be integrable on $G(I)$ for Lebesgue measure, it is necessary and sufficient that the function $t \mapsto f(G(t))g(t)$ be integrable on $I$ for Lebesgue measure, in which case the relation (13) holds.

The first part of the statement follows from the fact that the formula (12) is valid for every function $f \in \mathscr{K}(G(I))$; for, the support of the function $t \mapsto f(G(t))$ is contained in an interval $K \subset I$ on which $g$ is integrable, by virtue of the above remarks, and it suffices to apply Prop. 8 to $K$. The second part is a consequence of Th. 1 of No. 2.