Questions: Outer Measure and Carathéodory's Theorem

Question 1

An outer measure μ* assigns values to all subsets of ℝ. Which property distinguishes it from a genuine measure?

• A. It assigns ∞ to all unbounded sets
• B. It is only countably subadditive, not countably additive
• C. It is not monotone — larger sets can get smaller values
• D. It fails to assign 0 to the empty set

Answer: (B)

A genuine measure requires countable additivity: μ(⋃Aₙ) = Σμ(Aₙ) for disjoint measurable sets. An outer measure only satisfies countable SUBadditivity: μ*(⋃Aₙ) ≤ Σμ*(Aₙ). This weaker condition is exactly what allows μ* to be defined on all subsets of X — true countable additivity on all subsets of ℝ leads to contradiction (via Vitali's construction of a non-measurable set under the axiom of choice).

Question 2

A set E has the property that for every test set T ⊆ X, μ*(T) = μ*(T ∩ E) + μ*(T ∩ Eᶜ). What does Carathéodory's theorem conclude?

• A. E must be an open set, since only open sets split test sets cleanly
• B. E is Carathéodory-measurable, and the collection of all such sets forms a σ-algebra on which μ* is countably additive
• C. μ* is countably additive everywhere on X
• D. The infimum-of-coverings definition of μ*(E) agrees with its geometric length

Answer: (B)

The splitting property — μ*(T) = μ*(T ∩ E) + μ*(T ∩ Eᶜ) for every test set T — is Carathéodory's precise criterion for measurability. It captures the idea that E has a sharp enough boundary to partition any test set without loss. Remarkably, the collection of all sets satisfying this criterion automatically forms a σ-algebra, and restricting μ* to it yields a genuine countably additive measure — not just a subadditive function.

Question 3

The Carathéodory splitting property is motivated by the idea that a measurable set should not create ambiguity when used to partition any test set.

• T. True
• F. False

Answer: True

This is exactly the geometric intuition. If E is 'nicely shaped,' splitting any test set T into T ∩ E and T ∩ Eᶜ should not lose any measure — the pieces should add up exactly to μ*(T). Sets that fail this property have boundaries so irregular that they create measurement ambiguity. The splitting criterion is the precise algebraic encoding of this geometric idea.

Question 4

An outer measure is already a measure on most subsets of X, since it satisfies monotonicity, assigns 0 to the empty set, and is countably subadditive.

• T. True
• F. False

Answer: False

Outer measure satisfies three of the four measure axioms (non-negativity, μ*(∅) = 0, monotonicity), but it is only countably SUBadditive — not countably additive. Countable additivity is what distinguishes a genuine measure from an outer measure. Carathéodory's theorem solves exactly this problem: by restricting to measurable sets via the splitting property, the outer measure becomes a genuine measure on the resulting σ-algebra.

Question 5

Why can't we simply define Lebesgue measure on all subsets of ℝ, and how does Carathéodory's approach avoid this problem?

Model answer: Vitali's theorem shows that under the axiom of choice, there exist non-measurable subsets of ℝ that cannot be assigned a consistent countably additive measure. Carathéodory's approach sidesteps this by starting with an outer measure defined on all subsets (using only subadditivity), then using the splitting property to identify the measurable sets — those whose boundaries are sharp enough not to create measurement inconsistency. The resulting σ-algebra is vast (containing all open sets, closed sets, and their countable unions and intersections), but deliberately excludes pathological sets like the Vitali set.

The key insight is that Carathéodory doesn't assume which sets are measurable — the splitting property discovers them. This is why the construction works: you don't need to know in advance what the measurable sets are. The non-measurable sets are precisely those that fail the splitting property, and their existence is why naive measure-of-everything is impossible.