Exercise 3, § 7

The notations being the same as in Exer. 2, assume that X is an open subset of T. a) For every numerical function $g \geq 0$ defined on X, show that $\int^* g \, d\mu_X = \int_X^* g \, d\mu$ (use Exer. 2, Prop. 3 of §5 and Prop. 4 of §1). b) Let g be a function defined on X, with values in $\overline{\mathbf{R}}$ or in a Banach space, $g'$ the extension of g to T equal to 0 on $T - X$. For g to be $\mu_X$-integrable, it is necessary and sufficient that $g'$ be $\mu$-integrable, in which case $\int g \, d\mu_X = \int_X g \, d\mu$ (use a)).