Carathéodory Extension Theorem — Mathematics Knowledge System

Overview

The Carathéodory Extension Theorem provides a rigorous method to extend a pre-measure defined on an algebra (or semi-ring) of sets to a full countably additive measure on the generated sigma-algebra. It is the foundational result that justifies the construction of Lebesgue measure from the length function on intervals, and more generally allows measures to be defined via their values on simple sets and then extended uniquely to a complete measure space.

Intuition

You know how to measure 'simple' sets: the length of an interval is b - a. You want to measure complicated sets. The strategy: cover every set from the outside with countably many simple sets, and take the infimum of total lengths (this gives the outer measure). Then identify which sets behave additively (the measurable sets). The theorem says this process produces a genuine sigma-additive measure, and if the original pre-measure is sigma-finite, the extension is unique.

Formal Definition

Definition

Let A be an algebra of subsets of X and mu_0: A -> [0, inf] a pre-measure (finitely additive and countably additive when applicable). Define the outer measure mu*: P(X) -> [0, inf] by the covering formula. The Carathéodory Extension Theorem states that the restriction of mu* to the sigma-algebra M of mu*-measurable sets is a complete measure extending mu_0, and this extension is unique if mu_0 is sigma-finite.

\mu^*(E) = \inf\!\left\{ \sum_{n=1}^{\infty} \mu_0(A_n) : E \subseteq \bigcup_{n=1}^{\infty} A_n,\; A_n \in \mathcal{A} \right\}

Outer measure from pre-measure

\mathcal{M} = \{ E \subseteq X : \forall A \subseteq X,\; \mu^*(A) = \mu^*(A \cap E) + \mu^*(A \cap E^c) \}

Carathéodory-measurable sets

\bar{\mu} = \mu^*\big|_{\mathcal{M}} \text{ is a complete measure on } \mathcal{M} \supseteq \sigma(\mathcal{A})

The extension

Notation

Notation	Meaning
$\mathcal{A}$	Algebra on which the pre-measure is originally defined
$\mu_0$	Pre-measure on A
$\mu^*$	Outer measure induced by mu_0
$\bar{\mu}$	The complete measure on M (the extension of mu_0)

Proofs

Proof sketch: M is a sigma-algebra

$X \in \mathcal{M}$ (mu*(A ∩ X) + mu*(A ∩ X^c) = mu*(A) + mu*(empty) = mu*(A). So X satisfies Carathéodory's criterion.)
$E \in \mathcal{M} \Rightarrow E^c \in \mathcal{M}$ (The criterion is symmetric in E and E^c.)
$E_1, E_2 \in \mathcal{M} \Rightarrow E_1 \cup E_2 \in \mathcal{M}$ (Use both criteria for E_1 and E_2 applied successively to any test set A; algebra of equalities yields the union criterion.)
$\mathcal{M} \text{ is closed under countable unions}$ (By induction on finite unions and a limit argument using monotone convergence of outer measure.)

Theorems

Theorem 1: Carathéodory Extension Theorem

Let \mathcal{A} \text{ be an algebra on } X \text{ and } \mu_0 : \mathcal{A} \to [0,\infty] \text{ a pre-measure. Then there exists a measure } \mu \text{ on } \sigma(\mathcal{A}) \text{ with } \mu|_{\mathcal{A}} = \mu_0. \text{ If } \mu_0 \text{ is } \sigma\text{-finite, this extension is unique.}

Theorem 2: Uniqueness under sigma-finiteness

\text{If } \mu_0 \text{ is } \sigma\text{-finite (}X = \bigcup A_n,\; A_n \in \mathcal{A},\; \mu_0(A_n) < \infty\text{), then any two measures on } \sigma(\mathcal{A}) \text{ extending } \mu_0 \text{ are equal.}

Theorem 3: Completeness of the extension

\text{The measure space } (X, \mathcal{M}, \bar{\mu}) \text{ is complete: if } \bar{\mu}(N)=0 \text{ and } E \subseteq N, \text{ then } E \in \mathcal{M} \text{ and } \bar{\mu}(E) = 0.

Worked Examples

Let A = all finite unions of half-open intervals [a,b) in R. Define mu_0([a,b)) = b - a and extend additively to all of A.
$\mu_0\!\left(\bigsqcup_{k=1}^n [a_k, b_k)\right) = \sum_{k=1}^n (b_k - a_k)$
Verify mu_0 is a pre-measure: it is finitely additive by definition and sigma-additive on A (since a countable disjoint union in A that lands in A must be finite).
Apply the Extension Theorem: the induced outer measure lambda* is defined on all subsets of R, and its restriction to the Carathéodory-measurable sets M is a complete measure.
$\lambda^*(E) = \inf\!\left\{ \sum_n (b_n - a_n) : E \subseteq \bigcup_n [a_n, b_n) \right\}$
M contains sigma(A) = B(R), and since mu_0 is sigma-finite (R = union [-n,n)), the extension is unique. The resulting measure is Lebesgue measure lambda.

Answer: Lebesgue measure is the unique extension of the length pre-measure from A to sigma(A) = B(R), completed to the Lebesgue sigma-algebra M.

Practice Problems

Difficulty 8/10

Let A be an algebra and mu_0 a pre-measure. Show that the induced outer measure mu* satisfies mu*(A) = mu_0(A) for all A in A (i.e., mu* extends mu_0 to all of P(X)).

Difficulty 7/10

Explain why sigma-finiteness is needed for uniqueness. Give a concrete example where uniqueness fails.

Difficulty 6/10

What does the Carathéodory Extension Theorem guarantee when mu_0 is sigma-finite?

The extension to sigma(A) exists but may not be uniqueA unique complete measure on sigma(A) extending mu_0A unique measure on P(X) extending mu_0The extension exists only if A is already a sigma-algebra

Common Mistakes

Common Mistake

The extension always lands exactly on sigma(A).

The Carathéodory construction produces a complete measure on M, which generally contains sigma(A) strictly. The sigma-algebra M is the completion of sigma(A) with respect to the extended measure.

Common Mistake

Sigma-finiteness is always satisfied in practice.

Counting measure on an uncountable set is not sigma-finite. Many measures arising in functional analysis or topology are not sigma-finite, and uniqueness of extension is a non-trivial constraint.

Common Mistake

A pre-measure on an algebra is automatically sigma-additive.

A pre-measure is required to be sigma-additive for disjoint sequences whose union stays within the algebra. This is an additional hypothesis that must be verified, not a consequence of finite additivity alone.

Quiz

Historical Background

Constantin Carathéodory published his extension theorem in 1914, providing the definitive framework for constructing measures axiomatically. His approach via outer measures unified and generalised earlier work of Borel and Lebesgue, and remains the standard approach in modern measure theory textbooks.

1902
Lebesgue constructs measure on R by extending length from intervals
Henri Lebesgue
1914
Carathéodory publishes the general extension theorem via outer measures
Constantin Carathéodory
1950
Halmos's textbook codifies the theorem in the form used today
Paul Halmos

Summary

A pre-measure on an algebra A induces an outer measure on all subsets of X via the covering infimum construction.
Carathéodory's criterion identifies the measurable sets as those that split every test set additively.
These measurable sets form a complete sigma-algebra M ⊇ sigma(A) on which the outer measure is a genuine measure.
If the pre-measure is sigma-finite, the extension to sigma(A) is unique (Dynkin's theorem).
Lebesgue measure is the canonical application: length on half-open intervals extends uniquely to a complete measure on the Lebesgue sigma-algebra.

References

BookFolland, G. B. (1999). Real Analysis: Modern Techniques and Their Applications (2nd ed.). Wiley. Chapter 1.
BookHalmos, P. R. (1950). Measure Theory. Van Nostrand. Chapter 3.
WebsiteWikipedia — Carathéodory's extension theorem