Measure Theory 22 | Outer measures - Part 3: Proof [dark version]

A key step in turning an outer measure into a genuine measure is proving that the collection of “measurable” sets built from the outer measure forms a σ-algebra. Once that σ-algebra is in place, defining μ(A)=Φ(A) on it automatically yields a countably additive measure—exactly what measure theory needs for later results like Carathéodory’s extension theorem.

The proof starts by naming the σ-algebra candidate: all subsets A⊆X that satisfy the Carathéodory measurability condition relative to the outer measure Φ. Concretely, a set A is declared Φ-measurable if for every subset Q⊆X the outer measure splits additively across A and its complement: Φ(Q)=Φ(Q∩A)+Φ(Q∩Aᶜ). With that definition, the first σ-algebra checks are straightforward. The empty set is measurable because Q decomposes into Q∩∅ and Q∩X, and Φ(∅)=0 for any outer measure. Similarly, X is measurable since its complement is ∅, making the same additivity relation collapse in the same way.

Next comes closure under complements. If A is measurable, then for every Q the defining identity holds. Swapping A with Aᶜ in the identity and using that (Aᶜ)ᶜ=A recovers the same additivity formula for Aᶜ. That establishes that the measurable sets are closed under taking complements.

The hard part is closure under countable unions. The argument first proves closure under finite unions by taking two measurable sets A1 and A2 and showing A1∪A2 is measurable. The proof uses the measurability identities for A1 and A2 with a generic set Q, then combines them using subadditivity of Φ and a careful set-theoretic rewrite. A Venn-diagram style decomposition of intersections with Q, A1, and A2 shows that the inequality produced by subadditivity tightens to the exact additivity form required for measurability of A1∪A2.

For countably many sets, the proof assumes A1, A2, … are measurable and sets A=⋃_{j=1}^∞ A_j. A standard trick is applied: replace the sequence by a pairwise disjoint one without losing the union, which also aligns perfectly with how countable additivity is proved for measures. Using a generic set Q̂, the measurability condition is applied to the disjoint pieces so that intersections like Q̂∩A_j isolate each term. An induction establishes the finite-union identity: Φ(Q̂∩(A1∪…∪A_n)) equals the sum of Φ(Q̂∩A_j) for j=1…n.

To pass from finite to infinite unions, monotonicity of outer measures is used. As n increases, the complements shrink, letting the proof take limits and obtain an inequality involving Φ(Q̂∩A) and the infinite series Σ_{j=1}^∞ Φ(Q̂∩A_j). Finally, σ-subadditivity of Φ is invoked again to convert the series bound into an equality, forcing the Carathéodory additivity identity for A. Since Q̂ was arbitrary, this shows A is measurable, completing closure under countable unions.

As a payoff, the same identity with Q̂=A itself yields countable additivity for μ(A)=Φ(A) on the resulting σ-algebra. That completes the proposition’s Part A (σ-algebra construction) and Part B (μ becomes a measure), setting up the path to Carathéodory’s extension theorem in the next step.