Exercise 18, § 8

The notations being the same as in Exer. 17, let $\nu_\iota$ be a positive measure on $T_\iota$, of total mass equal to 1, and let $\nu = \bigotimes_{\iota \in I} \nu_\iota$.

a) Show that if one of the measures $\nu_\iota$ is alien to the measure $\mu_\iota$ with the same index, then $\nu$ is alien to $\mu$.

b) For every $\iota \in I$, write $\nu_\iota = \mu'_\iota + \mu''_\iota$, where $\mu'_\iota$ has base $\mu_\iota$, and $\mu''_\iota$ is alien to $\mu_\iota$ (§ 5, Th. 3); suppose that $\mu'_\iota \neq 0$ for all $\iota \in I$. Show that, for $\nu$ not to be alien to $\mu$, it is necessary and sufficient that $\mu''_\iota = 0$ except for a countable family of indices, and that the product $\mu' = \bigotimes_{\iota \in I} \mu'_\iota$ be a measure $\neq 0$ with base $\mu$; $\mu'' = \nu - \mu'$ is then alien to $\mu$. (If $\mu''_\iota \neq 0$ for an uncountable infinity of indices, show that there exists a number $\alpha$ such that $0 < \alpha < 1$ and such that, for a countable infinity of indices $\iota_n$, one has $\mu'_{\iota_n}(T_{\iota_n}) \leq \alpha$; show then that $\mu$ is alien to $\nu$ by using a); prove the second part of the proposition by using Exer. 17.)