Exercise 5, § 7

Let T and $\mu$ be the locally compact space and the measure defined in Exer. 5 of Ch. IV, §1, and let D be the (closed) set of points of T of the form $(0, y)$.

a) Show that the measure $\mu_D$ induced on D by $\mu$ is zero; however, $\int^* \varphi_D \, d\mu = +\infty$ (cf. Exer. 3 a)).

b) Let $X = T - D$, and let $j$ be the canonical injection of X into T; then $j(\mu_X) = \mu$, but $\int^*(\varphi_D \circ j) \, d\mu_X = 0$, $\int^* \varphi_D \, d\mu = +\infty$ (cf. Exer. 4).