Measure Theory 20 | Outer measures - Part 1 [dark version]

Outer measures provide a flexible way to assign “volumes” to *all* subsets of a space without requiring a sigma-algebra up front. Instead of starting with measurable sets, an outer measure is defined on the full power set of a fixed set X and satisfies three core properties: it assigns 0 to the empty set, it is monotone (if A ⊆ B then Φ(A) ≤ Φ(B)), and it is countably sub-additive (Φ(⋃_{n} A_n) ≤ Σ_{n} Φ(A_n)). This combination captures the intuition that unions can’t have more “size” than the sum of the sizes of their parts, even when overlaps occur.

The motivation matters because many standard measure-theoretic constructions fail on arbitrary subsets. In particular, countable additivity—where Φ(⋃ A_n) equals Σ Φ(A_n)—does not hold for general outer measures. The transcript highlights this by contrasting sub-additivity with the stronger “sigma additivity” property expected from a genuine measure. Since equality can break down on the whole power set, the path forward is to identify which subsets behave well enough for additivity to work.

That leads to the key next step: restricting attention to “good” sets, called Φ-measurable sets. A set A is Φ-measurable if it splits every other set Q into two parts—Q ∩ A and Q  A—such that Φ(Q) equals Φ(Q ∩ A) + Φ(Q  A). Geometrically, A acts like a boundary that doesn’t distort the outer-measure accounting when sets are decomposed. The transcript also notes a common variation in definitions across textbooks: some use an inequality direction (≥) instead of equality, but for outer measures the conditions end up equivalent.

The payoff comes in the form of a proposition: the collection of all Φ-measurable sets forms a sigma algebra. Once that structure is in place, the outer measure can be converted into an ordinary measure. Specifically, defining μ(A) = Φ(A) for measurable sets A produces a true measure on that sigma algebra. This is the foundation needed to prove major theorems later—especially those that rely on extending pre-measures—because it shows how outer-measure constructions can be turned into bona fide measures once the right sets are identified.

In short, outer measures start as a coarse tool defined everywhere, then Φ-measurability selects the subsets where additivity becomes exact. The resulting sigma algebra supports a genuine measure, setting up the technical machinery for the next stage of measure theory.