Exercise 4, § 4

Let $K$ be the discrete space consisting of the two numbers $0, 1$; let $A$ be an infinite set, and let $X$ be the product space $K^A$. Let

be an elementary set in $X$; if $h$ is the number of indices $\alpha$ such that $V_\alpha \neq K$, put $\mu(V) = 2^{-h}$.

a) Show that if $U_1, \ldots, U_n$ are pairwise disjoint elementary sets, then

[Write each $U_k$ in the form $W_k \times K^B$, where $B$ (independent of $k$) is the complement of a finite subset of $A$, and $W_k$ is a finite set; count up the number of points in $W_k$].

b) Deduce from a) that $X$ satisfies the condition $(D_{IV})$ of Exercise 7 of § 1. [If there were an uncountable family $(U_\lambda)$ of pairwise disjoint elementary sets, there would exist an integer $p > 0$ such that $\mu(U_\lambda) \geq 1/p$ for an infinite number of indices $\lambda$].

c) Show that if $A$ is uncountable, then $X$ does not satisfy the condition $(D_{III})$ of Exercise 7 of § 1 (cf. Exercise 3).