Exercise 16, § 6

Let $X$ be a metrizable space, $f$ a mapping from the set $\mathscr{R}(E)$ of compact subsets of $X$ into $\overline{\mathbf{R}}$, which satisfies condition (CA_1) (for two compact subsets of $X$) and is such that, for every decreasing sequence $(K_n)$ of compact subsets of $X$, one has

For every subset $A$ of $X$, $f_*(A)$ denotes the upper bound of the $f(K)$ for all compact subsets $K$ of $A$, and $f^*(A)$ the lower bound of the $f_*(U)$ for all open sets $U$ containing $A$. We say that $A$ is admissible for $f$ if $f_*(A) = f^*(A)$. a) Prove that for every compact subset $K$ of $X$ and every $\varepsilon > 0$, there exists an open set $U \supset K$ such that for every compact subset $L$ of $X$ such that $K \subset L \subset U$, one has (reason by contradiction). Deduce that every compact subset and every open subset of $X$ is admissible for $f$. b) Suppose that $f$ satisfies the following condition: (AL) For every pair of compact sets $K, K'$ of $X$, Prove that if $(K_i)$ is a finite family of compact sets such that $f\left( \bigcup_i K_i \right) < +\infty$, $(K'_i)$ a second finite family of compact sets having the same set of indices, such that $K'_i \subset K_i$ and $f(K'_i) > -\infty$ for every $i$, one has

EXERCISES

Let $(A_i), (B_i)$ be two finite families of subsets of $X$, having the same set of indices, such that $B_i \subset A_i$ for every $i$, and such that $f^*\left( \bigcup_i A_i \right) < +\infty$ and $f^*(B_i) > -\infty$ for every $i$. Prove that if $f$ satisfies (AL), one has

(Reduce to a set of indices with two elements. First consider the case where the $A_i$ and $B_i$ are open, and use the following lemma: if $U_1, U_2$ are two open sets, $K$ a compact set such that $K \subset U_1 \cup U_2$, there exist two compact sets $K_1 \subset U_1, K_2 \subset U_2$ such that $K \subset K_1 \cup K_2$.)

c) Suppose that $f$ satisfies condition (AL). Prove that for every increasing sequence $(A_n)$ of subsets of $X$ such that

one has $f^*\left( \bigcup_n A_n \right) = \sup_n f^*(A_n)$ (use $b$). Deduce that if $f$ does not take the value $-\infty$ in $\mathscr{R}(X)$, then $f^*$ is a right-continuous capacity on $X$.

d) Suppose that $f$ satisfies condition (AL). Prove that every union of a sequence $(A_n)$ of admissible sets such that $f^*(A_n) > -\infty$, is admissible. (Use $b$ and $c$).