Exercise 2, § 7

Let T be a locally compact space, X a locally compact subspace of T, and $\mu$ a positive measure on T; set $\nu = \varphi_X \cdot \mu = j(\mu_X)$ (j the canonical injection of X into T). a) Show that every $\mu_X$-negligible subset of X is $\nu$-negligible. b) For every numerical function $g \geq 0$ defined on X, show that $\int^* g \, d\mu_X = \int_X^* g \, d\nu$ (use a), as well as formula (7) of §3). c) 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 $\nu$-integrable, in which case $\int g \, d\mu_X = \int_X g \, d\nu$ (use b)).