Exercise 10, § 7

Let X, Y be two locally compact spaces. Show that every proper continuous mapping $\pi$ of X into Y is conservative. (The hypothesis implies that the mapping $f \mapsto f \circ \pi$ of $\mathscr{H}(Y)$ into $\mathscr{C}(X)$ has its image E contained in $\mathscr{H}(X)$; then, for every positive measure $\nu$ on Y, apply to the subspace E of $\mathscr{H}(X)$ and to the positive linear form $f \circ \pi \mapsto \nu(f)$ on E the Cor. of Prop. 1 of TVS, II, §3, No. 1.)