Proposition 5

Let f be a ν-measurable positive numerical function defined on X, and let N be the set of $t \in T$ such that f is not both $\lambda_t$-measurable and $\lambda_t$-moderated.

a) Suppose that f is ν-moderated. The set N is then locally μ-negligible, the function $t \mapsto \int^\bullet f(x) d\lambda_t(x)$ is μ-measurable, and

b) Suppose that f is ν-moderated, and that Λ is vaguely continuous. The set N is then μ-negligible, the function $t \mapsto \int^* f(x) d\lambda_t(x)$ is μ-measurable and μ-moderated, and

c) Suppose that $\lambda_t^\bullet(1) < +\infty$ locally μ-almost everywhere. The set N is then locally μ-negligible, the function $t \mapsto \int^\bullet f(x) d\lambda_t(x)$ is μ-measurable, and (9) holds.

Let us first prove a) (resp. b)), assuming that f is ν-moderated. The assertions concerning the set N have already been established (Prop. 4, and Cor. 2 of Prop. 3). By Prop. 6 of §1, No. 2, we may limit ourselves to proving a) (resp. b)) in each of the following special cases:

1) The function $f$ is $\nu$-negligible. 2) There exists a compact set $K$ such that $f$ is zero outside $K$ and the restriction of $f$ to $K$ is continuous.

The special case 1) has already been treated (Cor. 1 of Prop. 3). To treat the second, we denote by $G$ a $\nu$-integrable open set containing $K$, by $M$ a constant upper bound for $f$, by $h$ the lower semi-continuous function $M \varphi_G$, and by $g$ the function $h - f$. Since the function $f$ is upper semi-continuous on $X$, $g$ is lower semi-continuous and positive. Moreover, $f, g, h$ are $\nu$-integrable.

Let us then apply formula (4) (resp. (5)) to the lower semi-continuous functions $h$ and $g$. By subtraction, we see that the function

is $\mu$-measurable and that the formula (9) (resp. (10)) holds. Finally, under the hypothesis of b), the function $t \mapsto \int^* f(x)\, d\lambda_t(x)$ has finite upper integral, hence is indeed $\mu$-moderated.

To prove c), let us take up again the measures $\mu_\alpha$ and $\nu_\alpha$ of the proof of Prop. 3; since $f$ is $\nu_\alpha$-measurable and $\nu_\alpha$-moderated, the assertion a) implies that $t \mapsto \int^\bullet f(x)\, d\lambda_t(x)$ is $\mu_\alpha$-measurable and that

It remains only to sum on $\alpha$, applying Props. 1 and 2 of §2, No. 2.

If $f$ is not assumed to be $\nu$-moderated, and if one does not make the assumption in c), then the relation (9) may not hold (Exer. 3).