Theorem 1 (Prokhorov)

— *Let $\mathscr{T} = (T_i, p_{ij})$ be an inverse system of topological spaces indexed by $I$, $T$ a topological space and $(p_i)_{i \in I}$ a coherent and separating family of continuous mappings $p_i : T \to T_i$. Finally, let $(\mu_i)_{i \in I}$ be an inverse system of measures on $\mathscr{T}$.

For there to exist a bounded measure $\mu$ on $T$ such that $p_i(\mu) = \mu_i$ for all $i \in I$, it is necessary and sufficient that the following condition be satisfied:

(P) *for every $\varepsilon > 0$, there exists a compact subset $K$ of $T$ such that $\mu_i^\bullet(T_i - p_i(K)) \leq \varepsilon$ for all $i \in I$.

When this is so, the measure $\mu$ is uniquely determined and*

for every compact set $K$ in $T$.

Let us first prove the uniqueness of $\mu$. Let $\mu$ be a bounded measure on $T$ such that $p_i(\mu) = \mu_i$ for all $i \in I$. Let $K$ be a compact subset of $T$; by Prop. 2 of No. 1, the set $K$ is the intersection of the decreasing directed family $(\overline{p_i^{-1}(p_i(K))})_{i \in I}$ of closed subsets of $T$. By the Cor. of Prop. 5 of § 1, No. 6, we therefore have

which establishes the formula (2). Since two measures that coincide on the set of compact sets are equal (§ 1, No. 2, Cor. of Prop. 2), it follows that $\mu$ is unique.

By Prop. 3, there exists a bounded measure $\pi$ on $T$ such that $\pi^\bullet(K) = \inf_{i \in I} \mu_i^\bullet(p_i(K))$ for every compact subset $K$ of $T$. By formula (2), the existence of a bounded measure $\mu$ on $T$ such that $p_i(\mu) = \mu_i$ for all $i \in I$ is therefore equivalent to the relation:

(P') $p_i(\pi) = \mu_i$ for all $i \in I$.

For $i \leq j$, we have $\mu_i = p_{ij}(\mu_j)$, whence $\mu_i^\bullet(T_i) = \mu_j^\bullet(T_j)$; since $I$ is directed, there exists a finite number $c \geq 0$ such that $\mu_i^\bullet(T_i) = c$ for all $i \in I$. By Prop. 3, the measure $\mu_i - p_i(\pi)$ is positive, hence is zero if and only if its total mass is zero, that is, if $\mu_i(T_i) = p_i(\pi)^\bullet(T_i)$. Since $p_i(\pi)^\bullet(T_i) = \pi^\bullet(T)$, the condition (P') is thus equivalent to $\pi^\bullet(T) = c$, that is (§ 1, No. 2, Remark 3) to the property:

(P'') $\sup_{K \in \mathscr{K}} \pi^\bullet(K) = c$, where $\mathscr{K}$ is the set of compact subsets of $T$.

Now, for $K \in \mathscr{K}$, we have

and this formula immediately implies the equivalence of (P) and (P'').

Q.E.D.

Let $(T_i, p_{ij})$ be an inverse system of topological spaces. Set $T = \lim_{\leftarrow} T_i$ and denote by $p_i$ the canonical mapping of $T$ into $T_i$. Generalizing Def. 2 of Ch. III, § 4, No. 5, we shall say that a bounded measure $\mu$ on $T$ is the inverse limit of an inverse system $(\mu_i)_{i \in I}$ of measures if $\mu_i = p_i(\mu)$ for all $i \in I$. Th. 1 provides a criterion for the existence of inverse limits of measures. When the spaces $T_i$ are compact, and the mappings $p_{ij}$ surjective, $T$ is compact and $p_i(T) = T_i$ for every $i \in I$; the condition (P) is therefore fulfilled, and in this case we recover Prop. 8, (iv) of Ch. III, § 4, No. 5.