Exercise 4, § 7

The notations being those of Exer. 2, show that if X is closed in T then, for every numerical function $f \geq 0$ defined on T, $\int^*(f \circ j) \, d\mu_X = \int^* f \, d\nu$ (observe that the mapping j is proper, and use Prop. 2 of §4). Under the same conditions, if f is a mapping of T into $\overline{\mathbf{R}}$ or into a Banach space, for $f \circ j$ to be $\mu_X$-integrable it is necessary and sufficient that f be $\nu$-integrable, in which case $\int (f \circ j) \mu_X = \int f \, d\nu$ (§4, Th. 2).