Exercise 4, § 1

Let T and $\mu$ be the locally compact space and measure defined in Exer. 5 of Ch. IV, §1.

a) Show that $\mu$ is the supremum of a sequence of measures with finite support, but that $\mu^* \neq \mu^*$, and that $\mu$ is not moderated. From this, deduce that Prop. 11 becomes false when the symbol $\int^*$ therein is replaced by $\int^*$, even when the set A is countable.

b) For every real number $y$, let $f_y$ be the characteristic function of the set reduced to the point $(0, y)$ of T. Show that the mapping $t \mapsto (f_y(t))_{y \in \mathbf{R}}$ is $\mu$-measurable, but that if $f = \sup_{y \in \mathbf{R}} f_y$ then $\mu^*(f) = +\infty$ and $\sup_{y \in \mathbf{R}} \mu^*(f_y) = 0$.