Outer Measure and Carathéodory's Criterion
Length is obvious for intervals and hopeless in general. The way through is to define an over-estimate that works for every set, then find the tollgate that separates the honest sets from the rest.
In this subchapter
3.1.1What "length" must mean
The prize of Chapter 3 is a countably additive measure on the Borel sets of the line that assigns each interval its length — the completion of the promise made in 1.4 that at most one such measure exists. Before building it, fix the design constraints, because they explain every strange definition ahead.
Length must at least: give (a,b) the mass b − a; be countably additive on disjoint sets (that is the entire point — without it, "eventually" and "infinitely often" die, as 2.4 audited); and be defined on a σ-algebra containing the intervals. The Vitali construction of 0.3 says the domain cannot be everything: some subset must be left out. So the construction must do two opposite jobs at once — reach every set the rules of 1.2 force into the σ-algebra, and provably refuse the Vitali delegate V.
The strategy, in one sentence: first define a quantity for every subset with no exceptions (an over-estimate), then carve out of that lawless wilderness the sub-collection where additivity happens to hold — and prove the sub-collection is a σ-algebra containing the intervals.
3.1.2Outer measure: the cheapest over-estimate
For a set A ⊆ ℝ, cover it by a countable collection of open intervals (a₁,b₁), (a₂,b₂), … — meaning A ⊆ ⋃n (an, bn) — and add up the total interval length ∑n (bn − an). Then
the infimum — greatest lower bound, in plain words: the cheapest possible cover — over all countable covers. Two glosses. The infimum over an uncountable collection of covers exists because the sums are nonnegative (the set of possible totals has a bottom, 0, so it has a greatest lower bound). And allowing ∞: some sets cannot be covered by finite total length at all; the cheapest cover is then infinite and we write λ*(A) = ∞.
Take A = (a,b). Covers by intervals must pile up at least the length b − a: any countable cover of a compact sub-interval [a+ε, b−ε] has total length at least b − a − 2ε (a finite sub-collection already covers it, by compactness — flagged gloss: a closed bounded interval cannot be covered by intervals of total length less than its own), and letting ε → 0 gives λ*(A) ≥ b − a. The single-interval cover gives λ*(A) ≤ b − a. Equal — outer measure extends length on intervals, the design constraint met on day one. The same one-line cover shows λ*(ℝ) = ∞ and, pleasingly, that outer measure is shift-invariant: covering A and covering A + t cost the same, since shifting a cover is a cover of the same price.
Enumerate A = {x₁, x₂, …} and cover the n-th point by the interval (xn − ε2−n, xn + ε2−n) of length ε21−n. Total price: ∑n ε21−n = 2ε + ε + ε/2 + … ≤ 4ε — in fact the series telescopes to 4ε; the point is that it is finite and as small as we like. Let ε → 0: λ*(A) = 0. The rationals — dense, everywhere, topologically enormous — cost nothing to cover. Countable dust is invisible to length, no matter how it is scattered.
3.1.3The three properties, proved
λ* satisfies: (i) λ*(∅) = 0; (ii) monotonicity: A ⊆ B implies λ*(A) ≤ λ*(B); (iii) countable subadditivity: λ*(⋃n An) ≤ ∑n λ*(An). Note: subadditivity, and a price over-estimate — over-covers can overlap, and their overlaps are paid for twice.
(i) The empty set is covered by the empty collection, price 0. (ii) Any cover of B is a cover of A, so the cheapest cover of A costs no more — an infimum over a larger collection is smaller. (iii) Fix ε > 0. For each n, pick a cover of An of price at most λ*(An) + ε/2n — possible by definition of infimum: some cover is nearly cheapest. Pile all these countably many countable collections into one countable collection: it covers ⋃An, and its total price is at most
Let ε → 0. The trick — give the n-th set a budget of ε/2n slop so the total slop is ε, then exhaust ε — is the workhorse of measure theory; we will reuse it in 3.2 and in every convergence proof of Chapter 5. ∎
3.1.4Where additivity fails: the warning shot
Everything so far works for every subset. So the natural hope: maybe λ* is already countably additive on all of P(ℝ)? If it were, Chapter 3 would be two pages long and Vitali would have been wrong. He was not. The failure is visible as a shadow in this subchapter and is proved in full in 3.4.
Recall the Vitali machine of 0.3: the delegate set V ⊆ [0,1], and its rational shifts V + q — disjoint, covering [0,1], inside [−1,2]. If λ* were additive over this countable disjoint family, we would get
by squeezing — but shift-invariance (Example 1) makes every summand the same number p = λ*(V), and a countable sum of a constant p ≥ 0 is either 0 (if p = 0) or ∞ (if p > 0): neither can sit between 1 and 3. So additivity must fail somewhere in that machine — and since every other step is a theorem we have proved, the failure isolates V: either outer measure itself, or additivity over the V + q. 3.4 sharpens this shadow into a proof that V is unmeasurable. The lesson to carry: subadditivity's slack — the double-paid overlaps — is not sloppiness. It is the mathematical footprint of the sets that cannot be measured.
3.1.5Carathéodory's criterion: the tollgate
So λ* is defined on everything, additive on intervals, but subadditive in general. Which sets behave? Carathéodory's idea is one of those definitions that looks backwards at first and turns out to be exactly right: a set is good if it splits every other set's price correctly.
A set E ⊆ ℝ is (Lebesgue) measurable if for every test set T ⊆ ℝ:
The one inequality ≤ is automatic by subadditivity — that is the toll: E must never overestimate the price of a split. Measurability is the statement that E cuts every shop's bill exactly in two. The definition asks nothing about E's own price — it asks about E's behavior as a knife.
∅ is measurable: it cuts every bill into λ*(T) + 0. ℝ likewise. And the measurable sets are closed under shifts: if E passes the test for every T, then E + t passes for every T, by testing against T − t and using shift-invariance of λ* — the same mirror game as Theorem 1's proof in 1.4. Boring facts, but they are the screws that will hold the Vitali argument together in 3.4: whatever the tollgate rejects, it rejects in every shifted copy too.
The Vitali delegate V: if V were measurable, all its shifts would be too, and additivity over the disjoint countable family V + q would follow from the Carathéodory theorem below — resurrecting the impossible squeeze of the warning. So the tollgate must reject V. What is genuinely new in 3.4 is not the idea but the ledger: the criterion converts "no consistent length exists" into a specific, checkable failure of the knife property for one named set.
3.1.6The reward: a σ-algebra with a measure on it
The measurable sets form a σ-algebra ℒ ⊆ P(ℝ); it contains every interval (hence every Borel set); and λ = λ* restricted to ℒ is a countably additive measure with λ((a,b)) = b − a.
The full proof is a chain of four lemmas, each a finite computation with the knife definition, and it occupies the first half of 3.2 where each lemma gets its own block: closure under complement (one line from the symmetry of the criterion); additivity over two disjoint measurable sets (test against their union); closure under countable disjoint unions (induction plus the ε/2ⁿ trick); σ-algebra (disjointification, as in 1.4 Proposition 1). We do not defer silently: the proof is completed in the next subchapter, which exists precisely to carry it. ∎
A suspicion worth voicing: we defined measurability using λ*, which was defined using intervals — and now we claim intervals are measurable. Circularity? No: λ*'s definition uses intervals as covering material, a pricing choice; measurability is then a property of sets relative to that pricing. Proving intervals satisfy the knife property is real work (one of the lemmas in 3.2), not an assumption smuggled in. The design is exactly one revolution: crude prices for everything → knife criterion → exact prices for the well-behaved → the crude material itself is well-behaved.
Outer measure prices every set with no exceptions. Carathéodory's criterion is the tollgate: sets that cut every bill exactly are the ones length was meant for — and the Vitali set is the one that cannot pay.
We have the pricing system and the tollgate, and a statement of the grand theorem. Next subchapter: the four lemmas proved, the σ-algebra ℒ assembled, the measure λ certified countably additive, and the two loose ends of the course tied — existence of the uniform measure on [0,1] (closing the promise of 1.4) and completeness, the property that lets us sleep at night after Chapter 5's limit theorems.