Exercise 2, § 3

Let T and $\mu$ be the locally compact space and the measure $\geq 0$ defined in Exer. 5 of Ch. IV, §1. Take for X the set T equipped with the topology induced by that of $\mathbf{R}^2$, so that X is locally compact and countable at infinity. Let $t \mapsto \lambda_t$ be the mapping of T into $\mathscr{M}(X)$ such that $\lambda_t = \varepsilon_t$ for $t = (0, y)$, and $\lambda_t = \varepsilon_t / n^3$ for $t = (1/n, k/n^2)$. Show that the mapping $t \mapsto \lambda_t$ is $\mu$-adequate, vaguely $\mu$-measurable, not vaguely continuous, and that, if $\nu = \int \lambda_t d\mu(t)$ and $f = \varphi_I$, where I is the square in $\mathbf{R}^2$ with center 0 and sides 2, then