Constructing Lebesgue Measure on [0, 1]
The last subchapter priced every set and named the tollgate. This one walks the four gates in order — and at the end, the uniform measure on [0,1] exists, and with it the entire continuous side of probability.
In this subchapter
3.2.1The plan: four lemmas and a theorem
Everything reduces to the knife definition of 3.1: E is measurable if it splits every test set's outer price exactly, λ*(T) = λ*(T∩E) + λ*(T∖E). Let ℒ be the collection of measurable sets. The Carathéodory theorem claims: ℒ is a σ-algebra, intervals are in it, and λ = λ*|ℒ is countably additive. We prove it as four lemmas in sequence — each small, each reusable, and together they are the backbone of every measure ever constructed since (this is not just how Lebesgue measure is built; it is how every measure is built, including the product measures of 6.1 and the laws of stochastic processes in 8.3).
3.2.2The two-set lemmas
E ∈ ℒ implies Ec ∈ ℒ. And if E, F ∈ ℒ are disjoint, then E ∪ F ∈ ℒ and λ*(E ∪ F) = λ*(E) + λ*(F).
Complements: the criterion's right side is symmetric in E and Ec — T ∖ E = T ∩ Ec and T ∖ Ec = T ∩ E — so if E passes for every T, so does Ec. Disjoint pairs: let E, F be measurable and disjoint; we must show E ∪ F is measurable, i.e. for every T: λ*(T) = λ*(T ∩ (E∪F)) + λ*(T ∖ (E∪F)). Test E's measurability against T, then F's against the piece T ∩ E (legal — the criterion holds for every test set, this one included) and chain the splits: λ*(T) = λ*(T∩E) + λ*(T∖E), then split T∩E = (T∩E∩F) ∪ (T∩E∩Fc) — the first is empty by disjointness — giving λ*(T∩E) = λ*(T∩E∩F) + λ*(T∩E) = 0 + λ*(T∩E)… step carefully instead: split T ∖ E by the knife F: λ*(T∖E) = λ*((T∖E)∩F) + λ*((T∖E)∖F). Now T ∩ (E∪F) = (T∩E) ∪ ((T∖E)∩F), a disjoint union, and T ∖ (E∪F) = (T∖E) ∖ F. Assembling the two splits: λ*(T) = λ*(T∩E) + λ*((T∖E)∩F) + λ*(T∖(E∪F)), and the first two terms are subadditivity-exactly the outer price of the disjoint union T ∩ (E∪F) — for the additivity claim, apply the whole argument to T = E ∪ F itself: the criterion for E at T reads λ*(E∪F) = λ*(E) + λ*(F) since T∩E = E, T∖E = F. Both boxes ticked. ∎
The chain-of-splits maneuver in Lemma 1 — test one knife, then test the next knife on a piece — is the same fixed-slot trick as the π–λ proof of 1.4. Watch for it in Lemma 2: the countable case will be an induction that never "finishes", closed instead by continuity from below. That is the only new ingredient countability ever adds: an induction plus a limit, closed by 2.1.
3.2.3Countable additivity: the σ-algebra assembled
If E₁, E₂, … ∈ ℒ are pairwise disjoint, then ⋃n En ∈ ℒ and λ*(⋃n En) = ∑n λ*(En).
Finite stage: by Lemma 1 and induction, SN = E₁ ∪ … ∪ EN ∈ ℒ and λ*(SN) = ∑n≤N λ*(En). Limit stage: SN ↑ S = ⋃n En, so for any test set T, T ∖ SN ↓ T ∖ S, and monotonicity of outer measure gives λ*(T ∖ S) ≤ λ*(T ∖ SN). Now the knife at stage N: λ*(T) = λ*(T∩SN) + λ*(T∖SN) ≥ λ*(T∩SN) + λ*(T∖S), and since T ∩ SN ⊆ T ∩ S with measures increasing, letting N → ∞ along any subsequence gives λ*(T) ≥ λ*(T∩S) + λ*(T∖S) — wait, the limit of the left sides exists because λ*(T∩SN) is increasing and bounded; passing to the limit in an inequality between convergent sequences is legal. The reverse inequality is subadditivity, as always. So S ∈ ℒ. Additivity: apply the knife equality at T = S with the finite stage and let N grow — λ*(S) ≥ λ*(SN) = ∑n≤N λ*(En) for every N, so λ*(S) ≥ ∑n λ*(En), and subadditivity gives ≤. Equal. ∎
Complements: Lemma 1. Countable unions with overlaps: disjointify, exactly as in Proposition 1 of 1.4 — Dn = En ∖ (E₁ ∪ … ∪ En−1) is measurable (differences of measurable sets are measurable: E ∖ F = E ∩ Fc, and Lemma 1 gives intersections via De Morgan), the Dn are disjoint with the same union, and Lemma 2 lands the union in ℒ. ∎
3.2.4Intervals pass the tollgate
Every interval — open, closed, half-open, bounded or a half-line — belongs to ℒ. Consequently ℬ(ℝ) ⊆ ℒ.
By Lemma 3 it suffices to handle one shape and generate the rest: we take the half-line E = (−∞, 0]. Test set T, price p = λ*(T); we must show λ*(T) ≥ λ*(T∩E) + λ*(T∖E) (≤ is subadditivity). Fix ε > 0 and a cover of T by intervals of total price ≤ p + ε. Cut every covering interval at 0 into its left and right parts: each interval (a,b) splits into (a, 0]-ish and (0, b)-ish pieces whose lengths add back to exactly b − a (boundary overlaps of measure zero — use half-open pieces, whose prices telescope). The left pieces cover T∩E, the right pieces cover T∖E, and their total prices add to at most p + ε; so λ*(T∩E) + λ*(T∖E) ≤ p + ε. Let ε → 0. Since σ-algebra ℒ is closed under the operations, and the half-lines generate ℬ by Theorem 1 of 1.3, we get ℬ(ℝ) ⊆ ℒ. ∎
Lemmas 1–4, read in order, are precisely Carathéodory's theorem as stated in 3.1: ℒ is a σ-algebra containing ℬ(ℝ); λ = λ*|ℒ is countably additive (Lemma 2); intervals have their lengths (Example 1 of 3.1). Note what the construction bought beyond the minimum: ℒ is strictly larger than ℬ — it contains every set squeezed between Borel sets of equal price. That surplus is the subject of §6, and it is the difference between the Lebesgue σ-algebra and the Borel one.
3.2.5Existence at last: the uniform measure on [0,1]
Restrict to Ω = [0,1]: take F = ℒ ∩ [0,1] (the measurable subsets of the unit interval) and P(A) = λ(A). Then (Ω, F, P) is a probability space — the uniform measure on [0,1] — and it is the unique probability on ([0,1], ℬ) with P((a,b)) = b − a.
P(Ω) = λ([0,1]) = 1 (Lemma 4); countable additivity is Lemma 2 restricted; nonnegativity is inherited from the price. That is existence — the half that 1.4 Example 4 explicitly owed: the intervals-based uniqueness theorem there said at most one such measure on ℬ; we now exhibit one (on the larger ℒ, and its restriction to ℬ is the promised one, so both existence and uniqueness are delivered). The failure of 0.2 is now resolved from both ends: a uniform random real exists as a model — and the domain on which it lives cannot be all subsets, as 3.4 will finally prove. ∎
"Pick a uniform random real in [0,1]" is a certified sentence: the model ([0,1], ℒ, λ) exists, and events like "the real lies in the Cantor set" (probability 0 — 3.3), "its first decimal digit is 7" (probability 1/10 — the interval [0.7, 0.8)), or "it is rational" (probability 0 — countable dust, 3.1 Example 2) all have exact numbers. Every continuous distribution of practice inherits this: the normal, the exponential, any model with a density is this construction pushed through a change of variables — the machinery of 5.4.
3.2.6Completeness: the property that buys sleep
A measure space is complete if every subset of a null set is measurable (and hence null). ℒ is complete by construction: if λ(E) = 0 and A ⊆ E, then for any test set T, monotonicity gives λ*(T∩A) ≤ λ*(A) ≤ λ*(E) = 0, and the knife inequality for A follows — subsets of free sets are free.
Completeness is not automatic. On the Borel σ-algebra alone, there can exist a null Borel set with a subset that is not Borel (a consequence of the cardinality count in 1.3: Borel sets are only 𝔠 many, but a null Cantor set has 2^𝔠 subsets). Whether that subset is measurable depends on the σ-algebra: in ℒ it is (completeness), in ℬ it may fail to even be an event. This sounds like scholasticism until Chapter 5: limits of measurable functions are taken pointwise, and pointwise limits can drag you off the edge of ℬ into subsets-of-null territory. Completing the σ-algebra — closing it under subsets of null sets, as ℒ already is — is the insurance policy that every limit theorem keeps its output inside the domain. 3.3 shows the same care from the null-set side of the street.
Every measure ever built is this proof in disguise: price everything, tollgate the good knives, and certify the additivity you promised. Lebesgue measure is just the first instance — and the only one we ever build by hand.
The measure exists, is unique on the Borel sets, and its domain ℒ is complete. Before hunting the non-measurable set that proves the domain cannot be everything, we take inventory of the small: null sets — the sets of probability zero that are not the same as impossible — the Cantor set as their uncanny mascot, and completeness as the license to ignore them.