Exercise 14, § 6

Let $X$ be a metrizable space and let $f$ be a capacity on $X$ such that $f(A \cup B) \leq f(A) + f(B)$.

a) With the notation of Exercise 12, show that $f^*(A \cup B) \leq f^*(A) + f^*(B)$ and $f_*(A \cup B) \leq f_*(A) + f_*(B)$ for any two subsets $A, B$ of $X$. [If $K$ is a compact set contained in $A \cup B$, and $U$ is an open set containing $B$, write $K$ in the form $K = (K \cap U) \cup (K \cap \overline{U})$.]

b) Let $K$ be a compact subset of $X$ having the power of the continuum, and let $(A, B)$ be a partition of $K$ into two sets such that every compact subset of $A$ or of $B$ is countable [cf. § 5, Exercise 18 d)]. Show that if $f(\{x\}) = 0$ for all $x \in X$ and if $f(K) > 0$, then neither $A$ nor $B$ is capacitable with respect to $f$.