Exercise 2, § 4

Hypotheses and notations are those of Th. 1 of No. 2, for which we propose to indicate a new proof.

a) Let $K$ be a compact subset of $T$; for every $i \in I$, set $K_i = p_i(T)$, and denote by $q_i$ the mapping of $K$ onto $K_i$ that coincides on $K$ with $p_i$; for $i \leq j$, denote by $q_{ij}$ the mapping of $K_j$ into $K_i$ that coincides on $K_j$ with $p_{ij}$. Deduce from the preceding exercise the existence of a greatest positive measure $\pi^K$ on $K$ such that $q_i(\pi^K) = (\mu_i)_{K_i}$ for all $i \in I$.

b) If $K$ and $L$ are compact subsets of $T$ such that $K \subset L$, show that $(\pi^L)_K \geq \pi^K$; deduce from this that the set of measures of the form $i^K(\pi^K)$ (where $K$ runs over the set of compact subsets of $T$, and $i^K$ is the canonical injection of $K$ into $T$) admits a supremum $\mu$ in $\mathscr{M}_+(T)$.

c) Show that $p_i(\mu) = \mu_i$ for every $i \in I$ (it suffices to observe that $p_i(\mu) \leq \mu_i$ and that the measures $p_i(\mu)$ and $\mu_i$ have the same total mass by the hypothesis (P)).