Exercise 3, § 4

Let $I$ be the compact interval $[0, 1]$ of $\mathbf{R}$ and $F$ be the vector space $\mathscr{C}(I, \mathbf{R})$ of continuous real valued functions defined in $I$. Let $E$ be the product space $\mathbf{R}^F$; for all $a \in I$, let $\varepsilon_a$ be the element of $E$ such that $\varepsilon_a(f) = f(a)$ for all $f \in F$. a) Show that, when $x$ varies in $I$, the set $K$ formed by the $\varepsilon_x$ is compact in $E$. b) Let $\lambda$ be an element of $E$ such that $\lambda(f) = \int_0^1 f(t)\ dt$ for all $f \in F$ (Lebesgue measure). Show that, in $E$, $\lambda$ belongs to the closure of the convex envelope of $K$ but does not belong to this convex envelope (cf. FVR, II, p. 7, prop. 5).