Exercise 19, § 6

Let $\mu$ be Lebesgue measure in $\mathbf{R}$; define a capacity $f$ right continuous on $\mathbf{R}^2$, satisfying condition (AL) of Exercise 16 of IX, p. 122, by setting $f(A) = \mu^* (\mathrm{pr}_1 A)$.

a) Let $A$ be a bounded non-capacitable subset of $\mathbf{R}^2$ (Exercise 18), $B_0$ a circle $\|x\| = r$ such that A is contained in the open disk $\|x\| < r$, $B_1$ a circle $\|x\| = r'$ of radius $r' > r$. Now prove that $A \cup B_0$ and $A \cup B_1$ are capacitable though their intersection $A$ is not.

b) For every $n$, let $C_n$ be the set of the $x \in \mathbf{R}^2$ such that

Now prove that each of the sets $A \cup C_n$ is capacitable though their intersection is not, and that one has $\inf f(C_n) \neq f\left( \bigcap_n C_n \right)$*