Exercise 18, § 6

In Prop. 4 b), the conclusion may fail if one omits the hypothesis that $\pi'$ is continuous. (Take $T = \mathbf{R}$, $T' = \overline{\mathbf{R}}$, $T'' = \mathbf{R}$; take for $\mu$ Lebesgue measure, for $\pi$ the canonical injection; set $\pi'(x) = x$ for $x \in \mathbf{R}$ and $\pi'(\pm \infty) = 0$.)