Proposition 5

— Let $A$ be a subset of $T$; the following properties are equivalent: a) The set $A$ is contained in the union of a sequence of $\mu$*-integrable open sets*. b) The set $A$ is contained in the union of a sequence of $\mu$*-integrable sets*. c) The set $A$ is contained in the union of a sequence of compact sets and a $\mu$*-negligible set*.

It is clear that each of the properties a) and c) implies b). Conversely, b) implies a) because every set of finite outer measure is contained in an integrable open set (Ch. IV, §1, No. 4, Prop. 19), and b) implies c) because every integrable set is the union of a sequence of compact sets and a negligible set (Ch. IV, §4, No. 6, Cor. 2 of Th. 4).