Exercise 7, § 4

With the notations of Exercise 6, assume that the conditions of the exercise are satisfied. Show that for every function $f \in \mathscr{C}(X; \mathbf{C})$,

where the limit is taken with respect to the directed set of finite subsets of $L$, and where $x_J$ denotes $\operatorname{pr}_J(x)$ for every $x \in X$ and every $J \subset L$; similarly,

for every $x \in X$, provided that all of the measures $\mu_\lambda \ (\lambda \in L)$ have total mass 1 (where, for every subset $K$ of $L$, $\mu_K$ denotes the product of the subfamily of measures $(\mu_\lambda)_{\lambda \in K}$).