Exercise 2, § 7

Let $X$ be a compact space, $\mathscr{P}$ a set of lower semi-continuous numerical functions on $X$, taking values in $]-\infty, +\infty]$. Assume that $\mathscr{P}$ contains the finite constants; for every function $h \in \mathscr{P}$ and every positive measure $\mu$ on $X$, $\mu^*(h)$ is defined and $> -\infty$ (§4, Exer. 5).

a) For two positive measures $\mu, \nu$ on $X$, write $\mu \prec \nu$ if $\mu^*(h) \leq \nu^*(h)$ for every function $h \in \mathscr{P}$. Show that the relation $\mu \prec \nu$ is a pre-order relation on $\mathscr{M}_+(X)$; it implies that $\mu(1) = \nu(1)$, $c\mu \prec c\nu$ for all $c > 0$, and $\mu + \lambda \prec \nu + \lambda$ for every measure $\lambda \in \mathscr{M}_+(X)$.

b) Assume that $\mathscr{P} + \mathscr{P} \subset \mathscr{P}$ and that $c \cdot \mathscr{P} \subset \mathscr{P}$ for $c > 0$. Let $\mu$ be a positive measure on $X$, $f$ a function in $\mathscr{C}(X; \mathbf{R})$. Denote by $Q_f$ the set of functions $h \geq f$ belonging to $\mathscr{P}$, and by $M_\mu$ the set of measures $\lambda \in \mathscr{M}_+(X)$ such that $\lambda \prec \mu$. Show that

(For every function $g \in \mathscr{C}(X; \mathbf{R})$, let $p(g) = \inf_{h \in Q_g} \mu^*(h)$. Show that $p(g + g') \leq p(g) + p(g')$ and $p(cg) = c \cdot p(g)$ for $g, g'$ in $\mathscr{C}(X; \mathbf{R})$ and $c > 0$; prove on the other hand that $M_\mu$ is identical to the set of measures $\lambda \in \mathscr{M}_+(X)$ such that, for every function $g \in \mathscr{C}(X; \mathbf{R})$, one has $\lambda(g) \leq p(g)$, and conclude by applying the Hahn-Banach theorem.) If the functions in $\mathscr{P}$ are continuous and if $S_0(f)$ is the lower envelope of $Q_f$, then also $\mu(S_0(f)) = \sup_{\lambda \in M_\mu} \lambda(f)$.

c) If the set $\mathscr{P}_0$ of functions in $\mathscr{P}$ that are continuous and finite on X is total in $\mathscr{C}(X; \mathbf{R})$, then the relation $\mu \prec \nu$ is an order relation on $\mathscr{M}_+(X)$. If a measure $\nu \in \mathscr{M}_+(X)$ is maximal for this order relation, then so is every measure $\nu'$ such that $0 \leqslant \nu' \leqslant \nu$ (for the usual order) (argue by contradiction using a)). If $\mathscr{P}_0 = \mathscr{P}$ and $\mathscr{P}$ is total, then every increasing directed set for the order relation $\mu \prec \nu$ admits a supremum in $\mathscr{M}_+(X)$ for this relation; in particular, $\mathscr{M}_+(X)$ is inductive for this relation.

d) Suppose that the functions in $\mathscr{P}$ are continuous and finite, that $\mathscr{P}$ is total in $\mathscr{C}(X; \mathbf{R})$, and that $\mathscr{P} + \mathscr{P} \subset \mathscr{P}$ and $c \cdot \mathscr{P} \subset \mathscr{P}$ for all $c > 0$. For every function $f \in \mathscr{C}(X; \mathbf{R})$, denote by $S(f)$ the lower envelope of the functions $h \in -\mathscr{P}$ such that $f \leqslant h$; for every $\varepsilon > 0$, denote by $K_{f,\varepsilon}$ the set of $x \in X$ such that $S(f)(x) \geqslant f(x) + \varepsilon$. Show that the following conditions are equivalent: α) The measure $\mu \in \mathscr{M}_+(X)$ is maximal for the order relation $\lambda \prec \lambda'$. β) For every function $f \in \mathscr{C}(X; \mathbf{R})$, $\mu(S(f)) = \mu(f)$. β') For every function $f \in \mathscr{P}$, $\mu(S(f)) = \mu(f)$. γ) For every function $f \in \mathscr{C}(X; \mathbf{R})$ and every $\varepsilon > 0$, $\mu(K_{f,\varepsilon}) = 0$. γ') For every function $f \in \mathscr{P}$ and every $\varepsilon > 0$, $\mu(K_{f,\varepsilon}) = 0$. (Apply b) to $-\mathscr{P}$.)

e) Under the hypotheses of d), show that if $\mu$ and $\mu'$ are two positive measures on X that are maximal for the relation $\prec$, then so is $\mu + \mu'$. From this, deduce that the set of positive measures that are maximal for this order relation is a lattice for the usual order relation $\leqslant$.