Lebesgue measurability. Definition.
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.
With this in mind, we wish to choose a criteria which will create a restricted class of sets. The below definition is one way to choose the criteria. Note how it both acts as a criteria and as a useful property describing the limits/capabilities of Lebesgue measure for such sets.
Lebesgue measurability. Definition.
A set \( E \subset \mathbb{R}^d \) is said to be Lebesgue measurable iff for every real \( \varepsilon > 0 \) there exists exists an open set \( U \subset \mathbb{R}^d \) containing \( E \) such that \( m^*(U \setminus E) \le \varepsilon \). If \( E \) is Lebesgue measurable, we refer to \( m(E) := m^*(E) \) as the Lebesgue measure of \( E \).
Appeal to intuition
This definition hints at Littlewood's first principle—that measurable sets are almost open.
It is worth comparing Lebesgue and Jordan measure:
- Jordan measure
- Jordan measurable sets can be efficiently contained in elementary sets, with an error that has small Jordan outer measure.
- Lebesgue measure
- Lebesgue measurable sets can be efficiently contained in open, with an error that has small Lebesgue outer measure.
A later card covers other criteria for measurability which are equivalent to the above.