Exercise 16, § 8

Let $(T_n)$ be an infinite sequence of compact spaces, $\mu_n$ a positive measure on $T_n$ of total mass equal to 1, and $\mu$ the product measure $\bigotimes_{n=1}^\infty \mu_n$ on $T = \prod_{n=1}^\infty T_n$.

a) Let f be a function $\geqslant 0$, $\mu$-integrable on T. Let $(L_n)$ be an increasing sequence of subsets of $\mathbf{N}$, and, setting $M_n = \mathbf{N} - L_n$, let $g = \sup_n f_{L_n}$, $h = \sup_n f_{M_n}$ (the notations of Exer. 15). For every $\alpha > 0$, let $A_\alpha$ be the set of points $t \in T$ where $g(t) > \alpha$, and $B_\alpha$ the set of points $t \in T$ where $h(t) > \alpha$. Show that $\alpha \cdot \mu(A_\alpha) \leqslant \int f d\mu$ and $\alpha \cdot \mu(B_\alpha) \leqslant \int f d\mu$. (Observe that $A_\alpha$ is the set of $t \in T$ where at least one of the $f_{L_n}(t)$ is $> \alpha$, and express it as a countable union of sets $G_n$, pairwise disjoint and such that $\alpha \cdot \mu(G_n) \leqslant \int_{G_n} f d\mu$.)

b) Suppose that $(L_n)$ is an increasing sequence of finite subsets of $\mathbf{N}$, whose union is $\mathbf{N}$. Show that $f_{L_n}$ tends almost everywhere to f, and that $f_{M_n}$ tends almost everywhere in T to the constant $\int f d\mu$. (For every $\varepsilon > 0$, consider a continuous function g depending on only a finite number of variables and such that $\int |f - g| d\mu \leqslant \varepsilon$, and apply a) to the function $|f - g|$.)