Proposition 9

Suppose that all of the measures $\mu_\lambda$ are positive and that the family $(\mu_\lambda(1))_{\lambda \in L}$ is multipliable in $\mathbf{R}_+$ (with product possibly 0). Then there exists one and only one measure $\mu$ on $X$ such that, for every finite subset $J$ of $L$ and every function $f_J \in \mathscr{C}(X_J; \mathbf{C})$,

Moreover, the measure $\mu$ is positive and its total mass is given by

Let $F$ be the vector space consisting of the functions on $X$ of the form $f_J \circ \mathrm{pr}_J$, where $J$ runs over the directed set of finite subsets of $L$, and $f_J \in \mathscr{C}(X_J; \mathbf{C})$; we again say that $F$ is the space of continuous functions on $X$ that depend only on a finite number of variables. Lemma 3 of No. 5 shows that $F$ is dense in $\mathscr{C}(X; \mathbf{C})$, which proves the uniqueness assertion. If $\mu_{\lambda_0} = 0$ for some $\lambda_0 \in L$ then the measure $\mu = 0$ meets the requirements, since in the second member of (18) we have $\mu_J(f_J) = 0$ if $\lambda_0 \in J$ and $\prod_{\lambda \in L - J} \mu_\lambda(1) = 0$ if $\lambda_0 \notin J$. We can therefore suppose that $\mu_\lambda \neq 0$ for all $\lambda \in J$ and, since the measures $\mu_\lambda$ are positive, this implies that $\mu_\lambda(1) \neq 0$ for all $\lambda \in L$. Let us then set $\mu'_\lambda = \mu_\lambda / \mu_\lambda(1)$ for every $\lambda \in L$, so that $\mu'_\lambda$ is a positive measure on $X_\lambda$ such that $\mu'_\lambda(1) = 1$. It then follows from Prop. 8 of No. 5 that there exists a positive measure $\mu'$ on $X$ of total mass 1, such that $\mu'(f_J \circ \mathrm{pr}_J) = \mu'_J(f_J)$ for every finite subset $J$ of $L$ and every function $f_J \in \mathscr{C}(X_J; \mathbf{C})$. The positive measure

then meets the requirements, since

The measure $\mu$ defined in Prop. 9 is called the product measure of the family of positive measures $(\mu_\lambda)_{\lambda \in L}$ and is denoted by $\bigotimes_{\lambda \in L} \mu_\lambda$.