Exercise 12, § 6

Let $X$ be a metrizable space, and let $f$ be a mapping of the set $\mathscr{K}(X)$ of compact subsets of $X$ into $\overline{\mathbf{R}}$ which satisfies the conditions (CA$_I$) (for two compact subsets of $X$) and (CA$_{III}$). For each subset $A$ of $X$, let $f_*(A)$ denote the least upper bound of the numbers $f(K)$ for all compact subsets $K$ of $A$, and let $f^*(A)$ denote the greatest lower bound of the $f_*(U)$ for all open sets $U$ containing $A$. $A$ is said to be admissible with respect to $f$ if $f_*(A) = f_*(A)$.

a) Show that for every compact subset $L$ of $X$ and every $\varepsilon > 0$, there exists an open set $U \supset L$ such that, for every compact subset $K$ of $X$ for which $L \subset K \subset U$, we have $f(K) \leq f(L) + \varepsilon$. (Argue by contradiction.) Deduce that every compact subset and every open subset of $X$ is admissible with respect to $f$.

b) Suppose that $f$ satisfies the following condition:

(AL) For each pair of compact subsets $K, K'$ of $X$, we have

Show that if $(K_i)$ is a finite family of compact sets with the property that and if $(K'_i)$ is another finite family of compact sets indexed by the same set, such that $K'_i \subset K_i$ and $f(K'_i) > -\infty$ for each $i$, then we have

Let $(A_i), (B_i)$ be two finite families of subsets of $X$, with the same index set, such that $B_i \subset A_i$ for each $i$, $f^* \left( \bigcup_i A_i \right) < +\infty$ and $f^*(B_i) > -\infty$ for each $i$. Show that if $f$ satisfies (AL), we have

[Reduce to the case of an index set with two elements. Consider first the case where $A_i$ and $B_i$ are open sets, and use the following lemma: if $U_1, U_2$ are two open sets and $K$ is a compact set contained in $U_1 \cup U_2$, then there exist compact sets $K_i \subset U_i \ (i = 1, 2)$ such that $K \subset K_1 \cup K_2$.]

c) Suppose that $f$ satisfies condition (AL). Show that for every increasing sequence $(A_n)$ of subsets of $X$ such that $f^*(A_n) > -\infty$, we have $f^* \left( \bigcup_n A_n \right) = \sup_n f^*(A_n)$ [use b)]. Deduce that if $f$ does not take the value $-\infty$ on $\mathscr{K}(X)$, then $f^*$ is a capacity on $X$.

d) Suppose that $f$ satisfies the condition (AL). Show that the union of any sequence $(A_n)$ of admissible sets such that $f^*(A_n) > -\infty$ for each $n$ is admissible. [Use b) and c).]