Exercise 15, § 8

Let $(T_\iota)_{\iota \in I}$ be any family of compact spaces and, for each $\iota \in I$, let $\mu_\iota$ be a positive measure on $T_\iota$ of total mass 1; let $\mu$ be the product measure $\bigotimes_{\iota \in I} \mu_\iota$ on $T = \prod_{\iota \in I} T_\iota$. For every partition (L, M) of I into two subsets, one identifies T with the product $T_L \times T_M$, where $T_L = \prod_{\iota \in L} T_\iota$; for every $t \in T$, one writes $t_L = \mathrm{pr}_L t$, so that t is identified with $(t_L, t_M)$; let $\mu_L$ be the measure $\bigotimes_{\iota \in L} \mu_\iota$ on $T_L$. Let p be a finite real number $\geqslant 1$; for every function f in $\mathscr{L}_F^p(T, \mu)$ (F a Banach space or $F = \overline{\mathbf{R}}$), one writes $f_L(t) = \int f(t_L, t_M) d\mu_M(t_M)$. Show that, with respect to the directed set of finite subsets L of I, $f_L$ tends in mean of order p to f, and $f_M$ tends in mean of order p to the constant function equal to $\int f d\mu$. (Approximate f by a continuous function depending only on a finite number of variables.)

From this, deduce that if a $\mu$-measurable subset A of T is such that, for all $t \in A$, every point $t' \in T$, whose coordinates are equal to those of t except for a finite number of indices, also belongs to A, then $\mu(A) = 0$ or $\mu(A) = 1$.