Theorem 2

— Assume that the directed preordered set I has a countable cofinal subset. Let $\mathscr{T} = (T_i, p_{ij})$ be an inverse system of topological spaces, $T = \lim_{\leftarrow} T_i$ and $p_i$ the canonical mapping of $T$ into $T_i$. Then every inverse system $(\mu_i)_{i \in I}$ of measures on $\mathscr{T}$ admits an inverse limit.

We shall first treat the case that $I = \mathbf{N}$ and set $q_n = p_{n,n+1}$. Let $\varepsilon > 0$. Define recursively a sequence of compact sets $L_n \subset T_n$ as follows: $L_0$ is a compact subset of $T_0$ such that $\mu_0^\bullet(T_0 - L_0) \leq \varepsilon/2$, and for $n \geq 0$ the compact set $L_{n+1}$ is contained in $\overline{q_n^{-1}(L_n)}$ and satisfies

This construction is possible by virtue of Remark 3 of §1, No. 2. We have because $\mu_n = q_n(\mu_{n+1})$; by induction on $p$, one deduces that Since $T$ is a closed subspace of $\prod_{n \in \mathbf{N}} T_n$ and the product space $\prod_{n \in \mathbf{N}} L_n$ is compact, the subset $L = T \cap \prod_{n \in \mathbf{N}} L_n = \bigcap_{n \in \mathbf{N}} \overline{p_n^{-1}(L_n)}$ of $T$ is compact. Let $n \in \mathbf{N}$; we have $p_n(L) = \bigcap_{m \geq n} p_{nm}(L_m)$ (GT, I, §9, No. 6, Prop. 8) and $p_{nm}(L_m) \supset p_{nm'}(L_{m'})$ for $m' \geq m \geq n$, whence

But, for $m \geq n$, the measure $\mu_n$ is the image of $\mu_m$ under $p_{nm}$, whence

passing to the limit with respect to $m$, we obtain $\mu_n^\bullet(T_n - p_n(L)) \leq \varepsilon$. In other words, Prokhorov’s condition (P) is satisfied, and there exists a bounded measure $\mu$ on $T$ such that $\mu_n = p_n(\mu)$ for all $n \in \mathbf{N}$ (No. 2, Th. 1).

Let us pass to the general case: there exists in $I$ an increasing cofinal sequence $(i_n)_{n \in \mathbf{N}}$. The mapping $t \mapsto (p_{i_n}(t))_{n \in \mathbf{N}}$ is a homeomorphism of $T$ onto the inverse limit of the inverse system $(T_{i_n}, p_{i_n i_m})$ (GT, I, §4, No. 4). By the first part of the proof, there exists therefore a bounded measure $\mu$ on $T$ such that $\mu_{i_n} = p_{i_n}(\mu)$ for all $n \in \mathbf{N}$. Let $i \in I$; there exists an $n \in \mathbf{N}$ with $i \leq i_n$, whence

Q.E.D.

Theorem 2 is often used in the following situation: let $D$ be a countable set and $(X_t)_{t \in D}$ a family of topological spaces. Let $\mathfrak{F}$ be the set of finite subsets of $D$, ordered by inclusion. For $J$ in $\mathfrak{F}$, set $X_J = \prod_{t \in J} X_t$, and for $J \subset J'$ let $p_{JJ'}$ be the canonical projection of $X_{J'}$ onto the partial product $X_J$. Also set $X = \prod_{t \in D} X_t$ and denote by $p_J$ the canonical projection of $X$ onto the partial product $X_J$. One shows easily (cf. S, III, §7, No. 2, Remark 3) that the family $(p_J)_{J \in \mathfrak{F}}$ defines a homeomorphism of $X$ onto $\lim_{\leftarrow} X_J$. An inverse system of measures is then a family of bounded measures $\mu_J$ on $X_J$ such that $\mu_J = p_{JJ'}(\mu_{J'})$ for $J \subset J'$. There exists one and only one bounded measure $\mu$ on $X$ such that $\mu_J = p_J(\mu)$ for every finite subset $J$ of $D$ (Kolmogoroff’s theorem). One sometimes says that $\mu$ is the measure on $\prod_{t \in D} X_t$ having margins $\mu_J$.

In particular, suppose given, for every $t \in D$, a measure $\nu_t$ on $X_t$ of total mass 1. Set $\mu_J = \bigotimes_{t \in J} \nu_t$ for every finite subset $J$ of $D$. Let $J \subset J'$ be two finite subsets of $D$ and let $K = J' - J$; identifying $X_{J'}$ with $X_J \times X_K$, one has $\mu_{J'} = \mu_J \otimes \mu_K$, and since the measure $\mu_K$ has total mass 1, the projection of $\mu_J \otimes \mu_K$ on $X_J$ is equal to $\mu_J$. The measure on $X$ admitting the margins $\mu_J$ is denoted $\bigotimes_{t \in D} \nu_t$ and is called the product of the family $(\nu_t)_{t \in D}$. When the spaces $X_t$ are compact, we recover the construction of Ch. III, §4, No. 6.