Exercise 17, § 8

Let $(T_\iota)_{\iota \in I}$ be any family of compact spaces; for each $\iota \in I$, let $\mu_\iota$ be a positive measure on $T_\iota$, of total mass equal to 1, and let $\mu$ be the product measure $\bigotimes_{\iota \in I} \mu_\iota$ on $T = \prod_{\iota \in I} T_\iota$. For every $\iota \in I$, let $f_\iota$ be a function $\geqslant 0$ defined on $T_\iota$, $\mu_\iota$-integrable and such that $\int f_\iota d\mu_\iota \leqslant 1$; set $\mu'_\iota = f_\iota \cdot \mu_\iota$ and $\mu' = \bigotimes_{\iota \in I} \mu'_\iota$ (Ch. III, §4, Exer. 6). Assume that $\mu' \neq 0$.

a) For every finite subset J of I, set $T_J = \prod_{\iota \in J} T_\iota$, $\mu_J = \bigotimes_{\iota \in J} \mu_\iota$, $\mu'_J = \bigotimes_{\iota \in J} \mu'_\iota$, $f_J(t) = \prod_{\iota \in J} f_\iota(\mathrm{pr}_\iota t)$, $g_J = \sqrt{f_J}$, $\mu''_J = \sqrt{\mu_J \mu'_J}$ (§ 5, No. 9) and $\rho(\mu_J, \mu'_J) = \mu''_J(T_J) = \int g_J d\mu_J$. Show that, for the functions $g_J$ to converge in the quadratic mean to a function in $\mathscr{L}^2(T, \mu)$, with respect to the directed ordered set of finite subsets of $I$, it is necessary and sufficient that the product of the numbers $\rho(\mu_\iota, \mu'_\iota)$ be convergent in $\mathbf{R}_+^*$, that is, $> 0$. (For two finite subsets $J, L$ such that $J \subset L$, evaluate the norm $N_2(g_J - g_L)$ by means of the $\rho(\mu_\iota, \mu'_\iota)$.) Deduce from this that the functions $f_J$ then converge in mean to a function $f \geq 0$ such that $\mu' = f \cdot \mu$.

b) Show that if the product $\prod_{\iota \in I} \rho(\mu_\iota, \mu'_\iota)$ is zero, then the measures $\mu$ and $\mu'$ are alien (consider the set $A_J$ of points $t \in T$ such that $g_J(t) > 1$, and evaluate the measures $\mu(A_J)$ and $\mu'(T - A_J)$).