Measure Theory 21 | Outer measures - Part 2: Examples [dark version]

Outer measures are built to assign a “size” to every subset of a set X, even when exact additivity fails. An outer measure Φ maps the power set of X to nonnegative numbers (allowing ∞) and must satisfy three rules: Φ(∅)=0, monotonicity (A⊆B ⇒ Φ(A)≤Φ(B)), and σ-subadditivity (Φ(⋃n A_n) ≤ Σn Φ(A_n)). The core insight in this installment is how different constructions—some deliberately crude—still satisfy these outer-measure axioms, and how the most important example turns interval length into a set function via coverings and an infimum.

The first example takes X=ℝ and defines Φ(A)=0 if A is empty and Φ(A)=1 otherwise. This function immediately meets the outer-measure requirements: monotonicity holds because any nonempty set has value 1, and σ-subadditivity holds because the left side is either 0 or 1 while the right side is a sum of nonnegative terms that is at least 1 whenever any A_n is nonempty. Yet it fails to be a measure because σ-additivity breaks: disjoint nonempty sets don’t force the value to add up correctly.

A second example uses X=ℕ and defines Φ(A) as the cardinality of A when A is finite, and ∞ when A is infinite. Here the outer measure becomes the familiar counting measure. Unlike the first example, this one is actually σ-additive, so it is a genuine measure; integrating with respect to it reproduces ordinary sums and series.

The centerpiece is the construction of outer measure from one-dimensional Lebesgue length. Start with the length of bounded intervals I=(a,b), given by μ(I)=b−a. To extend beyond intervals, define Φ(A) for any A⊆ℝ by covering A with countably many intervals {I_j} and summing their lengths. Because the “best” cover may not be unique, Φ(A) is defined as the infimum of these total lengths over all countable interval coverings of A. This is why the construction is called “outer”: it approximates the size of A from the outside using measurable building blocks (intervals), then shrinks the approximation by taking the infimum.

The remainder of the lesson verifies that this Φ is truly an outer measure. Φ(∅)=0 follows by choosing empty intervals in the covering. Monotonicity is shown by noting that any interval cover of a larger set B automatically covers a subset A, so the infimum for A cannot exceed that for B. The σ-subadditivity proof is the technical heart: if any Φ(A_n)=∞, the inequality is automatic; otherwise, for each n one chooses interval covers whose total lengths are within an arbitrarily small ε_n of Φ(A_n). Using these covers, one builds a countable interval cover for ⋃n A_n, estimates Φ(⋃n A_n) by the total length of that cover, and then lets the ε_n error budget collapse to an arbitrarily small overall ε. The result is the exact inequality Φ(⋃n A_n) ≤ Σn Φ(A_n). The same covering-and-infimum strategy generalizes to higher-dimensional volume, setting up the path toward the n-dimensional Lebesgue measure.