Exercise 6, § 4

With the notations of No. 6 show that, for there to exist a measure $\mu \neq 0$ on $X$ such that

for every finite subset $J$ of $L$ and every function $f_J \in \mathscr{C}(X_J; \mathbf{C})$, it is necessary and sufficient that the following three conditions be satisfied:

$1^\circ$ $\mu_\lambda \neq 0$ for all $\lambda \in L$. $2^\circ$ There exists a finite subset $J_0$ of $L$ such that the family $(\mu_\lambda(1))_{\lambda \in L - J_0}$ is multipliable in $\mathbf{C}^*$ (hence has a product $\neq 0$). $3^\circ$ The family $(\|\mu_\lambda\|)_{\lambda \in L}$ is multipliable in $\mathbf{R}_+^*$ (hence has a product $\neq 0$).

Under what conditions is the second member of (1) equal to 0 for every finite subset $J$ of $L$ and every function $f_J \in \mathscr{C}(X_J; \mathbf{C})$?