Exercise 1, § 7

a) Let $K$ be a compact convex subset of a Hausdorff locally convex space $E$, $\mu$ a positive measure on $K$ of mass 1, $b_\mu$ its barycenter. Show that for every positive function $f$ on $K$ that is concave and lower semi-continuous, one has $f(b_\mu) \geq \int^* f d\mu$. (Note that $f$ is bounded (TVS, II, §2, Exer. 32), consequently $-f$ is $\mu$-integrable.) From this, conclude that if $f$ is a lower semi-continuous (or upper semi-continuous) function on $K$ that is both convex and concave, then $\int f d\mu = f(b_\mu)$.

b) Let $I$ be the interval $[0,1]$ of $\mathbf{R}$, $K$ the subset of $\mathscr{M}_+(I)$ formed by the positive measures of total mass 1, which is convex and compact for the vague topology, and let $j : x \mapsto \varepsilon_x$ be the canonical injection of $I$ into $K$, which is a homeomorphism of $I$ onto a subspace of $K$. For every measure $\nu \in K$, set $g(\nu) = \sum_{x \in I} \nu(\{x\})$; this is a function that is both convex and concave in $K$; moreover, if $g_n(\nu)$ is the supremum of the $\nu(A)$ for all the finite subsets $A$ of $I$ having at most $n$ elements, then $g_n$ is upper semi-continuous on $K$, and $g(\nu) = \lim_{n \to \infty} g_n(\nu)$ for every $\nu \in K$, therefore $g$ is $\lambda$-integrable for every measure $\lambda$ on $K$. Let $\mu$ be the measure on $K$ such that $\int f d\mu = \int_I f(j(x)) dx$ for $f \in \mathscr{K}(X; \mathbf{R})$, which is positive and of total mass 1; show that $b_\mu$ is Lebesgue measure on $I$ and that $\int g d\mu \neq g(b_\mu)$.