Exercise 5, § 1

For every $t \in \mathbf{R}$ let $f(t)$ be the continuous function $x \mapsto \exp(-2\pi i tx)$, which is an element of the space $\mathscr{C} = \mathscr{C}_C(\mathbf{R})$ of continuous complex functions on $\mathbf{R}$. The space $\mathscr{C}$ is in duality with the space $\mathscr{C}' = \mathscr{C}'_C(\mathbf{R})$ of complex measures on $\mathbf{R}$ with compact support; the space $\mathscr{K} = \mathscr{K}_C(\mathbf{R})$ of continuous complex functions with compact support may be canonically identified with a subspace of $\mathscr{C}'$, by identifying a function $\varphi \in \mathscr{K}$ with the measure having density $\varphi$ with respect to Lebesgue measure. One denotes by $\mathscr{D}$ the subspace of $\mathscr{K}$ formed by the indefinitely differentiable complex functions with compact support. Show that, when $\mathscr{C}$ is equipped with the topology $\sigma(\mathscr{C}, \mathscr{C}')$ or with the topology $\sigma(\mathscr{C}, \mathscr{K})$, $f$ is not scalarly integrable for the Lebesgue measure $\mu$; however, for the topology $\sigma(\mathscr{C}, \mathscr{D})$, $f$ is scalarly $\mu$-integrable and $\int f d\mu$ is the measure $\varepsilon_0$; prove that in this case, the conditions of Prop. 7 are verified.*