Exercise 13, § 8

Let $(T_\iota)_{\iota \in I}$ be a family of compact spaces, and for each $\iota \in I$, let $\mu_\iota$ be a positive measure on $T_\iota$ of total mass equal to 1. Let $T$ be the product space $\prod_{\iota \in I} T_\iota$, and $\mu$ the product measure $\bigotimes_{\iota \in I} \mu_\iota$ on $T$ (Ch. III, §4, No. 5).

a) For every $\iota \in I$, let $K_\iota$ be a compact subset of $T_\iota$; show that if $K = \prod_{\iota \in I} K_\iota$ then $\mu(K) = \prod_{\iota \in I} \mu_\iota(K_\iota)$.

b) Let $M$ be a compact subset of $T$. Show that there exists a countable subset $J$ of $I$ such that, setting $H = I - J$, $T_J = \prod_{\iota \in J} T_\iota$, $\mu_J = \bigotimes_{\iota \in J} \mu_\iota$, $T_H = \prod_{\iota \in H} T_\iota$, one has $M \subset N \times T_H$, where $N$ is a $\mu_J$-measurable subset of $T_J$ such that $\mu_J(N) = \mu(M)$. (Observe that for every $n > 0$, there exists an open neighborhood of $M$, the union of a finite number of elementary sets, whose measure differs from $\mu(M)$ by less than $1/n$.) From this, deduce that for almost every $x \in N$, the section $M(x) \subset T_H$ (when $T$ is identified with the product $T_J \times T_H$) contains the product of the supports of the measures $\mu_\iota$ for $\iota \in H$ (make use of the Lebesgue–Fubini theorem).

c) Show that if $I$ is countable and if, for every $\iota \in I$, $A_\iota$ is a $\mu_\iota$-measurable subset of $T_\iota$, then the set $A = \prod_{\iota \in I} A_\iota$ is $\mu$-measurable and $\mu(A) = \prod_{\iota \in I} \mu_\iota(A_\iota)$.

d) Suppose that $I$ is uncountable. For every $\iota \in I$, let $A_\iota$ be a $\mu_\iota$-measurable subset of $T_\iota$. For $A = \prod_{\iota \in I} A_\iota$ to be $\mu$-measurable, it is necessary and sufficient that one be in one of the following two cases: $1^\circ \prod_{\iota \in I} \mu_\iota(A_\iota) = 0$; $2^\circ$ $A_\iota$ contains the support of the measure $\mu_\iota$, except perhaps for the indices $\iota$ of a countable subset of $I$. In each of these two cases, $\mu(A) = \prod_{\iota \in I} \mu_\iota(A_\iota)$. (Assuming that one is not in either of the two preceding cases, show that one can reduce to the case that $\mu_\iota(A_\iota) = 1$ for all $\iota \in I$. Using $b$), show that neither $A$ nor $T - A$ can contain a compact set of nonzero measure.)