Exercise 18, § 6

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

a) With the notations of exerc. 16, of IX, p. 122, now prove that one has

and

for any two subsets $A, B$ of $X$. (If $K$ is a compact set such that $K \subset A \cup B$, $U$ an open set such that $B \subset U$, write $K = (K \cap U) \cup (K \cap \complement U)$.)

b) Let $K$ be a compact subset of $X$ having the power of the continuum, $(A, B)$ a partition of $K$ into two sets such that every compact subset of $A$ or of $B$ is countable (cf. IX, p. 114, exerc. 17 d)). Now prove that if $f(\{x\}) = 0$ for every subset reduced to a point and if $f(K) > 0$, neither $A$ nor $B$ are capacitable for $f$.