Proposition 8

(i) If an inverse system $(\mu_\alpha)$ of measures on the $X_\alpha$ has an inverse limit, then that limit is unique. (ii) If an inverse system $(\mu_\alpha)$ has an inverse limit, then the family of norms $(\|\mu_\alpha\|)$ is bounded. (iii) If the $p_{\alpha\beta}$ are surjective and the family $(\|\mu_\alpha\|)$ is bounded, then the inverse system of measures $(\mu_\alpha)$ has an inverse limit.

(iv) *If the $p_{\alpha \beta}$ are surjective, then every inverse system $(\mu_\alpha)$ of positive measures has an inverse limit $\mu$, which is a positive measure on $X$, and $\| \mu \| = \| \mu_\alpha \|$ for all $\alpha$.

(i) We first prove the following lemma: