Exercise 21, § 6

Let T, X, Y be three locally compact spaces, $\pi : T \to X$, $\pi' : X \to Y$ universally measurable mappings. Then $\pi' \circ \pi$ is universally measurable. (Let $\mu$ be a positive measure on T. To prove that $\pi' \circ \pi$ is $\mu$-measurable, reduce to the case that T is compact and $\pi$ is continuous, and use the fact that $\pi'$ is measurable for $\pi(\mu)$.)