Theorem 2

Let $(\pi, g)$ be a $\mu$-adapted pair, and let

Let $f$ be a function defined on $X$, with values in a Banach space $F$ or in $\overline{\mathbf{R}}$. For $f$ to be essentially $\nu$-integrable, it is necessary and sufficient that $t \mapsto f(\pi(t))g(t)$ be essentially $\mu$-integrable, in which case Suppose, moreover, that $\pi$ is continuous and proper, and that $g$ is continuous and such that $g(t) > 0$ for all $t \in T$. Then, for $f$ to be $\nu$-integrable, it is necessary and sufficient that $t \mapsto f(\pi(t))g(t)$ be $\mu$-integrable.

A) We begin by treating the case that the measure $\mu$ has compact support $K$, on which $g$ is bounded. The measures $\mu$ and $\nu$ are then bounded, and one can replace 'essentially integrable' in the statement by 'integrable'. Suppose that $f$ is $\nu$-integrable: the function $f(\pi(t))g(t)$ is then $\mu$-integrable, and the relation (9) is verified, by Th. 1 of §3, No. 3. Conversely, suppose that $f(\pi(t))g(t)$ is $\mu$-integrable: $f$ is then $\nu$-measurable (No. 3, Prop. 3 and Remark), and

(No. 2, Th. 1); $f$ is therefore essentially $\nu$-integrable (§1, No. 3, Prop. 9), hence $\nu$-integrable. Th. 1 of §3, No. 3 then implies (9).

B) Let us pass to the general case. Let $\mathscr{K}$ be the set of compact subsets $K$ of $T$ such that $g|K$ is continuous: $\mathscr{K}$ is $\mu$-dense (Ch. IV, §5, No. 10, Prop. 15), therefore the measure $\mu$ is the sum of a family $(\mu_\alpha)_{\alpha \in A}$ of measures whose supports are elements of $\mathscr{K}$ (§2, No. 3, Prop. 4). The pair $(g, \pi)$ is obviously $\mu_\alpha$-adapted for every $\alpha \in A$, and the measure $\nu$ is the sum of the family of measures $\nu_\alpha = \int \varepsilon_{\pi(t)}g(t) \, d\mu_\alpha(t)$ (§3, No. 1, Cor. of Prop. 1). Since the argument of A) may be applied to the measures $\mu_\alpha, \nu_\alpha$, the first part of the statement then follows from Prop. 3 of §2, No. 2.

For the function $f$ (resp. $t \mapsto f(\pi(t))g(t)$) to be integrable for $\nu$ (resp. for $\mu$), it is necessary and sufficient that it be essentially integrable and that

The second part of the statement therefore follows from the first part and Proposition 2.