Example

— Let $(X_\lambda)_{\lambda \in L}$ be a family of compact spaces; set $X = \prod_{\lambda \in L} X_\lambda$ and, for every finite subset $J$ of $L$, set $X_J = \prod_{\lambda \in J} X_\lambda$; denote by $\mathrm{pr}_J : X \to X_J$ and $\mathrm{pr}_{J, K} : X_K \to X_J$ (for $J \subset K$) the canonical projections. We know that $(X_J, \mathrm{pr}_{JK})$ is an inverse system of compact spaces, and that the inverse limit of the system of continuous mappings $(\mathrm{pr}_J)$ is a homomorphism of $X$ onto the inverse limit space $\lim_{\leftarrow} X_J$, permitting one to identify these two spaces (GT, I, §4, No. 4 and S, III, §7, No. 2, Remark 3). Since the projections $\mathrm{pr}_{J, K}$ are surjective, it follows from Prop. 8 that the set $\mathscr{M}(X; \mathbf{C})$ (resp. $\mathscr{M}_+(X)$) may be identified with the set of inverse systems $(\mu_J)$ such that the family of norms $(\| \mu_J \|)$ is bounded (resp. such that the $\mu_J$ are all positive, and necessarily of the same total mass).

Let us consider in particular the case where, for each $\lambda \in L$, a measure $\mu_\lambda$ is given on $X_\lambda$ and one sets $\mu_J = \bigotimes_{\lambda \in J} \mu_\lambda$. If $J \subset K$ are two finite subsets of $I$ then, for every function $f_J \in \mathscr{C}(X_J; \mathbf{C})$ we have, by virtue of formula (14) of No. 4,

For $(\mu_J)$ to be an inverse system of measures, it is therefore necessary and sufficient that either $\mu_\lambda = 0$ for all $\lambda \in L$ or $\mu_\lambda(1) = 1$ for all $\lambda \in L$.