Exercise 1, § 4

Let $\mathscr{T} = (K_i, p_{ij})$ be an inverse system of compact spaces indexed by the set $I$, let $K$ be a compact space, and let $(p_i)_{i \in I}$ be a coherent and separating family of continuous mappings $p_i : K \to K_i$.

(2) A cardinal $c$ is said to be strongly inaccessible if it is uncountable, $2^b < c$ for every cardinal $b < c$, and $\sum_{i \in I} c_i < c$ for every family of cardinals $c_i < c$ such that $\mathrm{Card}(I) < c$. Exercises 11 and 13 show that, assuming the continuum hypothesis, the smallest cardinal that is not an Ulam cardinal is strongly inaccessible. It is not known whether the axiom of the existence of strongly inaccessible cardinals is contradictory to the other axioms of set theory.

a) Let $A$ be the set of continuous functions on $K$ of the form $f_i \circ p_i$, where $i$ runs over $I$ and $f_i$ runs over the set of continuous real functions on $K_i$. Show that $A$ is a dense linear subspace of $\mathscr{C}(K)$.

b) Let $(\mu_i)_{i \in I}$ be a sub-inverse system of positive measures on $\mathscr{T}$. Assume that the $p_i$ are surjective. For every $f \in A$, let $I_f$ be the set of $i \in I$ such that $f$ is of the form $f_i \circ p_i$ with $f_i \in \mathscr{C}(K_i)$ (necessarily unique). Show that there exists one and only one measure $\pi$ on $K$ such that $\pi(f) = \inf_{i \in I_f} \mu_i(f_i)$ for all $f \in A$. Show that $\pi$ is the greatest of the positive measures $\mu$ on $K$ such that $p_i(\mu) \leq \mu_i$ for all $i \in I$.