Exercise 19, § 6

Let $T, X, Y$ be locally compact spaces, $\pi : T \to X$, $\pi' : X \to Y$ mappings, $\pi'' = \pi' \circ \pi$, $\mu$ a real measure on $T$. It can happen that $\pi$ is $\mu$-proper and $\pi'$ is $\pi(\mu)$-proper, without $\pi''$ being $\mu$-proper. (Take $T = \mathbf{R} \times \{0, 1\}$, $X = \mathbf{R}$, $Y = \{0\}$, $\pi = \mathrm{pr}_1$; take for $\mu$ the measure that induces Lebesgue measure (resp. the opposite of Lebesgue measure) on $\mathbf{R} \times \{0\}$ (resp. $\mathbf{R} \times \{1\}$) canonically identified with $\mathbf{R}$.)