Exercise 4, § 3

With the spaces T, T', X and the measures $\mu, \mu', \nu$ having the same meaning as in Exer. 3, consider the product space $Y = T \times X = T \times (T \times T')$, and on Y the product measure $\varpi = \mu \otimes \nu$. If, for $t \in T$, one sets $\rho_t = \varepsilon_t \otimes \nu$, one has $\varpi = \int \rho_t d\mu(t)$. Let H be a subset of T that is not measurable for $\mu$ (Ch. IV, §4, Exer. 8), and let A be the subset of Y formed by the points $(t_1, t_2, t_3)$ satisfying the conditions $t_1 = t_3$, $t_2 \in H$. Show that the characteristic function $\varphi_A$ is locally negligible for $\varpi$, but that $\varphi_A$ is not $\rho_t$-measurable for any value of $t \in T$.