Remark 1

The condition 4) may be replaced, in the statement of Theorem 1, by the following condition ('right-continuity'): $4')$ For every $K \in \mathscr{K}(T)$ and every $\varepsilon > 0$, there exists an open set $U$ containing $K$, such that $I(H) \leq I(K) + \varepsilon$ for every compact set $H \subset U$. For, if $\mu$ is a measure, the function $I : K \mapsto \mu^*(K)$ satisfies $4'$ (§ 1, No. 9, Prop. 13). Conversely, suppose that $I$ satisfies 1) and $4'$; let us show that $I$ then satisfies 4). With notations as in the statement of Theorem 1, choose an $\varepsilon > 0$ and an open set $U$ containing the compact set $K = \bigcap_{\alpha \in A} K_\alpha$ and such that $4'$) is satisfied. There then exists an index $\beta \in A$ such that $K_\beta \subset U$, and this implies

and 4) is indeed verified.