Exercise 1, § 6

Let T and X be two locally compact spaces, $\mu$ a positive measure on T, $\pi$ a continuous and $\mu$-proper mapping of T into X, and let $\nu = \pi(\mu)$.

a) Show that for every numerical function $g \geqslant 0$, defined and lower semi-continuous on X, $g \circ \pi$ is lower semi-continuous on T and $\nu^*(g) = \mu^*(g \circ \pi)$.

b) Show that if f is a $\nu$-integrable function with values in a Banach space, then $f \circ \pi$ is $\mu$-integrable and $\int f d\nu = \int (f \circ \pi) d\mu$. Is the converse valid without a supplementary condition (cf. §4, Exer. 2b))?