Exercise 1, § 3

Let T be the compact interval [0, 1] of $\mathbf{R}$, and let $\mu$ be Lebesgue measure on T. Let X be the locally compact space obtained by equipping the interval [0, 1] with the discrete topology. The mapping $t \mapsto \lambda_t = \varepsilon_t$ of T into $\mathscr{M}(X)$ is scalarly essentially integrable, with integral $\nu = 0$. For the constant function f equal to 1 on X, the formula (6) of Prop. 3 fails. From this, deduce that the mapping $t \mapsto \lambda_t$ is not $\mu$-adequate.