Exercise 1, § 8

Let T, T' be two locally compact spaces, $\mu$ a positive measure on T, $\mu'$ a positive measure on T'. Show that if the measure $\mu'$ is bounded, then the projection $\mathrm{pr}_1$ of $T \times T'$ onto T is a $(\mu \otimes \mu')$-proper mapping and $\mathrm{pr}_1(\mu \otimes \mu') = a \cdot \mu$, where $a = \mu'(T')$. Deduce from this an example of a $(\mu \otimes \mu')$-negligible subset of $T \times T'$ whose projection on T is not $\mu$-measurable.