Exercise 9, § 7

Let X, Y be two locally compact spaces, $\pi : X \to Y$ a continuous mapping. The mapping $\pi$ is said to be conservative if, for every positive measure $\nu$ on Y that is concentrated on $\pi(X)$, there exists a positive measure $\mu$ on X such that $\nu = \pi(\mu)$.

a) Let X, Y, Z be three locally compact spaces, $\pi : X \to Y$ and $\pi' : Y \to Z$ two continuous mappings; set $\pi'' = \pi' \circ \pi$ and assume that the mapping $\pi$ is surjective. Show that if $\pi$ and $\pi'$ are conservative, then so is $\pi''$; conversely, if $\pi''$ is conservative then so is $\pi'$.

b) Assume that for every $y \in \pi(X)$ there exists an open neighborhood V of y in Y such that, if $\pi_V : \pi^{-1}(V) \to V$ is the mapping that coincides with $\pi$, there exists a continuous section associated with $\pi_V$ (GT, I, §3, No. 5). Show that $\pi$ is conservative (make use of Exer. 8).

c) Let Y be the interval $[0, 1]$ of $\mathbf{R}$, X the set Y equipped with the discrete topology, and $\pi : X \to Y$ the identity mapping. Show that the mapping $\pi$ is not conservative.