Exercise 40, § 7

Let $X$ be a compact convex set in $E$, a Hausdorff locally convex space, and let $f$ be an upper semi-continuous function that is bounded below. Let $g$ be a lower semi-continuous concave function, such that $g \geq f$. Show (with that notations of exerc. 38) that $g \geq \tilde{f}$. (Reduce to the case where $\inf_{x \in X} (g(x) - f(x)) > 0$. If $(f_\alpha)$ is a decreasing directed family of continuous functions such that $f = \inf(f_\alpha)$, show that then also $\inf_{x \in X} (g(x) - f_\alpha(x)) \geq 0$ for $\alpha \geq \alpha_0$, and so reduce the problem to the case where $f$ is continuous. Then use exerc. 38, e).)