Exercise 5, § 4

Let $(X_\lambda)_{\lambda \in L}$ be an infinite family of compact spaces; for each $\lambda \in L$, let $a_\lambda$ be a point of $X_\lambda$ and let $\mu_\lambda$ be a positive measure on $X_\lambda$ of total mass 1. For every finite subset $J$ of $L$, let $\nu(J)$ be the measure on $X = \prod_{\lambda \in L} X_\lambda$ that is the product of the measures $\mu_\lambda$ for $\lambda \in J$ and the measures $\varepsilon_{a_\lambda}$ for $\lambda \in L - J$. Show that, with respect to the directed set of finite subsets of $L$, the measure $\nu(J)$ tends vaguely to $\mu = \bigotimes_{\lambda \in L} \mu_\lambda$ but does not tend strongly to $\mu$.