Construct the Lebesgue measure from outer measure via Carathéodory's criterion
The length of an interval [a, b] is simply b − a. But what is the "length" of an arbitrary subset of ℝ — say, the rationals in [0, 1], or a fractal like the Cantor set? The Lebesgue measure extends the intuitive notion of length to a vast class of subsets of ℝ, assigning each one a non-negative real number (or +∞) in a way that is consistent with interval length, translation-invariant, and countably additive.
The construction of Lebesgue measure proceeds in two stages. First, we define outer measure, which provides an upper bound on the "size" of any subset by covering it with countably many intervals. Then we use Carathéodory's criterion to select those sets that split every other set cleanly into two parts — these are declared measurable, and on them the outer measure becomes a true measure.
This two-stage approach — approximate from outside, then filter for well-behaved sets — is one of the most elegant constructions in all of analysis. It produces the Lebesgue σ-algebra, which is strictly richer than the Borel σ-algebra, and the Lebesgue measure, which is the completion of Borel measure on ℝ.
The Lebesgue outer measure of a set A ⊆ ℝ is defined as m*(A) = inf { Σₙ ℓ(Iₙ) : A ⊆ ∪ₙ Iₙ, each Iₙ an open interval }, where ℓ(Iₙ) denotes the length of the interval Iₙ. In words, we cover A by countably many open intervals and sum their lengths; the outer measure is the infimum over all such coverings. This guarantees m*(A) ≤ the "true size" we would want A to have.
Outer measure is defined for every subset of ℝ, but it is not countably additive on all subsets — only countably subadditive: m*(∪ₙ Aₙ) ≤ Σₙ m*(Aₙ). This subadditivity is the fundamental limitation that forces us to restrict to a σ-algebra of measurable sets if we want a true (countably additive) measure.
Key properties of outer measure include: m*(∅) = 0, monotonicity (A ⊆ B implies m*(A) ≤ m*(B)), and the fact that m*([a, b]) = b − a for every interval. Outer measure also assigns measure zero to every countable set — for instance, m*(ℚ) = 0 even though the rationals are dense in ℝ.
The demo below illustrates how outer measure works by covering sets with intervals. Adjust the covering to see how the total length of the intervals provides an upper bound, and observe how tighter coverings yield a better approximation of the true measure.
Key insight: Outer measure always overestimates (or equals) the true size of a set because coverings can overlap. The infimum over all coverings squeezes this overestimate down to the correct value for measurable sets. For non-measurable sets, outer measure is the best we can do.
A set E ⊆ ℝ is Lebesgue measurable (in the sense of Carathéodory) if for every test set A ⊆ ℝ, m*(A) = m*(A ∩ E) + m*(A ∩ Eᶜ). Intuitively, E is measurable if it "splits" every set cleanly: the outer measure of any set A is exactly the sum of the outer measures of the part inside E and the part outside E.
This condition may look strange at first, but it is remarkably powerful. It automatically guarantees that the collection of measurable sets forms a σ-algebra, and that outer measure restricted to this σ-algebra is countably additive. The beauty of Carathéodory's approach is its generality: the same criterion works for constructing measures in any abstract setting, not just on ℝ.
Every Borel set satisfies Carathéodory's criterion, so B(ℝ) ⊆ L (the Lebesgue σ-algebra). But L is strictly larger: it contains all subsets of measure-zero sets, making Lebesgue measure a complete measure. This completeness is one of the key advantages of Lebesgue measure over Borel measure.
Countable additivity: If E₁, E₂, … are pairwise disjoint measurable sets, then m(∪ₙ Eₙ) = Σₙ m(Eₙ). This is the defining property that makes Lebesgue measure a true measure. It allows us to compute the measure of complicated sets by decomposing them into simpler pieces.
Translation invariance: For any measurable set E and any real number t, m(E + t) = m(E), where E + t = {x + t : x ∈ E}. This captures the geometric intuition that shifting a set does not change its size. Translation invariance, combined with countable additivity and the normalization m([0, 1]) = 1, actually characterizes Lebesgue measure uniquely.
Regularity: Lebesgue measure is both inner regular and outer regular. For any measurable set E, m(E) = inf {m(U) : E ⊆ U, U open} = sup {m(K) : K ⊆ E, K compact}. This means we can approximate any measurable set from outside by open sets and from inside by compact sets, bridging the abstract definition with concrete geometric intuition.