We want to describe a class of sets such that the Lebesgue outer measure obeys nice properties. Every subset of $$\mathrm{R}^d$$ has a Lebesgue outer measure (possibly infinite). Unfortunately, it turns out that if we include all sets, we lose properties like the union of some disjoint sets has measure which is the sum of the individual sets.
A set $$E \subset \mathbb{R}^d$$ is said to be Lebesgue measurable iff [for every ? there exists ? such that ?]. If $$E$$ is Lebesgue measurable, we refer to $$m(E) := m^*(E)$$ as the Lebesgue measure of $$E$$.