Exercise 15, § 6

Let $\mu$ be the Lebesgue measure on $\mathbf{R}$; define a capacity $f$ on $\mathbf{R}^2$ satisfying the condition (AL) of Exercise 12 by setting $f(A) = \mu^*(\mathrm{pr}_1A)$ (Proposition 15).

a) Let $A$ be a non-capacitable bounded subset of $\mathbf{R}^2$ (Exercise 14), let $B_0$ be a circle $||x|| = r$ such that $A$ is contained in the open disc $||x|| < r$, and let $B_1$ be a circle $||x|| = r'$ of radius $r' > r$. Show that $A \cup B_0$ and $A \cup B_1$ are capacitable, although their intersection $A$ is not.

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

Show that each of the sets $A \cup C_n$ is capacitable, although their intersection is not, and that we have $\inf_n f(C_n) \neq f\left( \bigcap_n C_n \right)$. \*