Corollary 1

— For a set $A$ to be integrable, it is necessary and sufficient that, for every $\varepsilon > 0$, there exist a compact set $K \subset A$ such that $|\mu|^*(A - K) \leq \varepsilon$. The measure $|\mu|(A)$ is then the supremum of the set of measures $|\mu|(K)$ of the compact sets $K \subset A$.

The condition is necessary, because if $G$ and $K$ satisfy the conditions of Theorem 4, then $|\mu|^*(A - K) \leq |\mu|^*(G - K) \leq \varepsilon$.

The condition is sufficient, because it says that, for the topology of convergence in mean, $\varphi_A$ is in the closure of the set of integrable functions $\varphi_K$ ($K$ an arbitrary compact subset of $A$).