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Carathéodory's extension theorem

Carathéodory's extension theorem is a cornerstone of measure theory, stating that given a premeasure \mu_0 defined on a ring \mathcal{R} of subsets of a set X, there exists a unique extension to a measure \mu on the \sigma-algebra \sigma(\mathcal{R}) generated by \mathcal{R}, provided \mu_0 is \sigma-finite.^[1] The theorem, introduced by Constantin Carathéodory in his 1914 paper "Über das lineare Maß von Punktmengen — eine Verallgemeinerung des Längenbegriffs," constructs this extension by first defining an outer measure \mu^* on the power set of X as the infimum of sums of \mu_0 over countable covers from \mathcal{R}, then identifying the Carathéodory-measurable sets—those E \subseteq X satisfying \mu^*(A) = \mu^*(A \cap E) + \mu^*(A \setminus E) for all A \subseteq X—which form a \sigma-algebra containing \mathcal{R} on which \mu^* restricts to a complete, countably additive measure agreeing with \mu_0.^[2]^[1] This result provides a systematic method to construct measures beyond elementary ones, with the outer measure ensuring subadditivity while the measurability condition guarantees additivity on the extended \sigma-algebra.^[3] Uniqueness holds when the premeasure is \sigma-finite, meaning X can be covered by countably many sets of finite \mu_0-measure, preventing non-unique extensions that can arise in infinite cases.^[1] The theorem's significance lies in its role in defining the Lebesgue measure on \mathbb{R}^n, where the premeasure is the elementary content on intervals, extended uniquely to the Lebesgue \sigma-algebra.^[3] It also generalizes to more abstract settings, such as Hausdorff measures in geometric measure theory, and underpins the construction of product measures via extensions from rectangle algebras.^[1] Carathéodory's approach, building on Lebesgue's earlier ideas but providing a more rigorous axiomatic framework, resolved foundational issues in integration and probability by ensuring measures are defined on rich \sigma-algebras while maintaining desirable properties like completeness.^[2]

Introduction

Overview

Carathéodory's extension theorem is a cornerstone of measure theory, enabling the construction of measures on sigma-algebras generated by simpler collections of sets. It addresses the challenge of extending a premeasure—defined on a semi-ring of subsets—to a full measure that preserves additivity properties over countable unions, thereby facilitating the development of Lebesgue measure and other fundamental constructions in analysis.^[4] The theorem asserts that if μ is a σ-finite premeasure on a semi-ring S of subsets of a set X, then there exists a unique measure on the sigma-algebra σ(S) generated by S that extends μ. Here, σ-finiteness ensures that X can be covered by countably many sets of finite μ-measure, which is crucial for uniqueness. This extension guarantees that the resulting measure is countably additive on σ(S).^[4]^[3] Central to the theorem is Carathéodory's method of defining an outer measure from the premeasure and identifying measurable sets as those satisfying a specific splitting condition, which yields the desired sigma-algebra and measure. This approach, originally developed by Constantin Carathéodory, underpins much of modern integration theory.^[5]^[4]

Historical context

The development of measure theory in the early 20th century emerged from efforts to provide a rigorous foundation for integration and the analysis of point sets, addressing shortcomings in classical approaches like the Riemann integral. Émile Borel laid initial groundwork with his 1898 work Leçons sur la théorie des fonctions, where he introduced the concept of measure for certain classes of sets in the real line, focusing on Borel sets generated by intervals to handle continuity and measurability in function theory.^[6] This was part of broader attempts to quantify "size" for non-denumerable sets, influenced by Cantor's set theory and the need for tools in analysis.^[6] Henri Lebesgue advanced this framework significantly in his 1902 thesis Intégrale, longueur, aire, published in the Annales de la Faculté des sciences de Toulouse. There, Lebesgue defined outer measure as the infimum of coverings by intervals, enabling the construction of Lebesgue measure on a sigma-algebra of measurable sets and providing a more general integration theory that encompassed discontinuous functions. However, Lebesgue's approach was tailored primarily to Euclidean spaces and relied on specific coverings, leaving open the challenge of systematically extending measures from smaller collections of sets, such as algebras, to full sigma-algebras while preserving countable additivity. Constantin Carathéodory addressed these limitations in his 1914 paper Über das lineare Maß von Punktmengen — eine Verallgemeinerung des Längenbegriffs, published in the Nachrichten von der Gesellschaft der Wissenschaften zu Göttingen. Building directly on Lebesgue's outer measure ideas, Carathéodory generalized the notion of length to arbitrary point sets in metric spaces by defining a premeasure on elementary families (like semi-rings of sets) and extending it via outer measures constructed from countable covers by these elementary sets.^[2] His theorem provided a general method to extend such premeasures to sigma-finite measures on the generated sigma-algebra, resolving issues of uniqueness and completeness in prior constructions and solidifying the axiomatic structure of measure theory.^[2] This work marked a pivotal evolution, shifting focus from ad hoc definitions to a unified extension principle applicable beyond Lebesgue's Euclidean context.^[7]

Preliminaries

Semi-rings and rings

In measure theory, a semi-ring of sets is a fundamental algebraic structure used to build measures from simpler collections of subsets. Let X be a set and \mathcal{S} \subseteq \mathcal{P}(X) a collection of subsets of X. Then \mathcal{S} is a semi-ring if it satisfies the following properties: the empty set \emptyset \in \mathcal{S}; it is closed under finite intersections, meaning that for all A, B \in \mathcal{S}, A \cap B \in \mathcal{S}; and for all A, B \in \mathcal{S} with A \subseteq B, the difference B \setminus A can be expressed as a finite disjoint union of sets from \mathcal{S}, i.e., there exist C_1, \dots, C_n \in \mathcal{S} with C_i \cap C_j = \emptyset for i \neq j such that B \setminus A = \bigcup_{i=1}^n C_i.^[8]^[9] A ring of sets extends the semi-ring structure by ensuring closure under additional operations. Specifically, a ring \mathcal{R} \subseteq \mathcal{P}(X) is a non-empty collection that contains \emptyset, is closed under finite unions (so if A, B \in \mathcal{R}, then A \cup B \in \mathcal{R}), and is closed under set differences (so if A, B \in \mathcal{R}, then A \setminus B \in \mathcal{R}). Equivalently, rings are closed under symmetric differences and finite intersections, and they automatically include finite disjoint unions. Every ring is a semi-ring, since the difference property follows from closure under unions and differences.^[8]^[10] The connection between semi-rings and rings lies in generation: given a semi-ring \mathcal{S}, the ring it generates, consisting of all finite disjoint unions of sets from \mathcal{S}. This construction ensures that any premeasure defined on \mathcal{S} can be extended uniquely to the generated ring.^[11]^[12] A classic example arises on the real line \mathbb{R}, where the collection of all half-open intervals of the form [a, b) with a, b \in \mathbb{R} and a < b, together with \emptyset, forms a semi-ring. The intersections of such intervals remain half-open intervals or empty, and differences like [a, b) \setminus [c, d) decompose into at most two disjoint half-open intervals. The ring generated by this semi-ring comprises all finite disjoint unions of such intervals, which include more general bounded sets with finite length.^[8]^[13]

Premeasures

In measure theory, a premeasure is a set function defined on a semi-ring that satisfies certain additivity properties, serving as the foundational structure for extending to a full measure via Carathéodory's theorem.^[14] Specifically, given a semi-ring S of subsets of a space X, a premeasure \mu: S \to [0, \infty] is a function such that \mu(\emptyset) = 0, and it exhibits countable additivity whenever applicable within the semi-ring.^[14] The additivity condition requires that for any countable collection of pairwise disjoint sets \{A_k\}_{k=1}^\infty \subseteq S whose union A = \bigcup_{k=1}^\infty A_k also belongs to S, the premeasure satisfies

\mu(A) = \sum_{k=1}^\infty \mu(A_k).

^[14] This implies finite additivity as a special case, where for finitely many disjoint sets A_1, \dots, A_n \in S with \bigcup_{i=1}^n A_i \in S, \mu\left( \bigcup_{i=1}^n A_i \right) = \sum_{i=1}^n \mu(A_i).^[14] Such additivity ensures consistency for unions that remain within the semi-ring, without requiring closure under arbitrary countable unions. Unlike a measure, which is defined on a \sigma-algebra and countably additive for all disjoint countable unions within the domain, a premeasure is restricted to a semi-ring and only demands additivity for those countable disjoint unions that happen to lie in the semi-ring itself.^[14] This limitation reflects the semi-ring's structure, which is closed under finite intersections and differences but not necessarily under complements or infinite unions, making premeasures a preliminary tool for construction rather than a complete measure space.^[14] A premeasure \mu on S is said to be \sigma-finite if the underlying space X can be expressed as a countable union X = \bigcup_{j=1}^\infty X_j, where each X_j \in S and \mu(X_j) < \infty.^[14] This condition plays a crucial role in ensuring uniqueness properties during the extension process, though it is not part of the basic definition of a premeasure.^[14]

Key properties

One fundamental motivation for employing semi-rings in the construction of measures arises from their suitability for defining simple, intuitive premeasures on basic sets, such as the length function on half-open intervals in \mathbb{R}, while recognizing their limitations in capturing the full structure required for integration over sigma-algebras like the Borel sets. These structures enable the initial specification of a premeasure on a collection that is closed under finite intersections and where differences can be expressed in a controlled manner, paving the way for extension to the larger sigma-algebra generated by the semi-ring via Carathéodory's theorem.^[15] A key approximation property of semi-rings ensures that for any two sets A, B in the semi-ring with B \subseteq A, the difference A \setminus B can be expressed as a finite disjoint union of sets from the semi-ring. This property facilitates the generation of the ring from the semi-ring, as every element of the generated ring can thus be represented as a finite disjoint union of semi-ring sets, allowing premeasures defined on the semi-ring to be straightforwardly extended to the ring while preserving additivity.^[15] If a premeasure on a semi-ring satisfies countable additivity—meaning that for any countable collection of pairwise disjoint sets in the semi-ring whose union is also in the semi-ring, the premeasure of the union equals the sum of the premeasures—it automatically induces finite additivity on the generated ring. This follows directly from the approximation property, as finite additivity on the ring is a consequence of applying countable additivity to the finite disjoint decompositions into semi-ring elements.^[16] Finally, every ring of subsets generates a sigma-algebra (or more precisely, a sigma-ring, which becomes a sigma-algebra if the ambient space is included) as its smallest extension closed under countable unions and complements. This generated sigma-algebra can be constructed either as the monotone class generated by the ring, leveraging the monotone class theorem, or via transfinite induction over the ordinals up to the first uncountable ordinal.^[17]

The theorem

Formal statement

Carathéodory's extension theorem states that if X is a set and \mathcal{S} is a semi-ring of subsets of X, then any \sigma-finite premeasure \mu on \mathcal{S} (that is, a countably additive set function \mu: \mathcal{S} \to [0, \infty] with \mu(\emptyset) = 0) can be extended to a measure on the \sigma-algebra generated by \mathcal{S}.^[2]^[1] The outer measure \mu^* induced by \mu is defined for every subset A \subseteq X by

\mu^*(A) = \inf\left\{ \sum_{i=1}^\infty \mu(A_i) : A_i \in \mathcal{S},\ A \subseteq \bigcup_{i=1}^\infty A_i \right\},

where the infimum is taken to be +\infty if no such cover exists and $0 if A = \emptyset.^[3]^[1] A subset E \subseteq X is Carathéodory measurable if for every test set T \subseteq X,

\mu^*(T) = \mu^*(T \cap E) + \mu^*(T \cap E^c).

The collection of all Carathéodory measurable sets forms a σ-algebra containing σ(𝒮) on which the restriction of μ* is a complete measure that extends μ (agreeing with μ on 𝒮), and under the σ-finiteness assumption on μ (meaning X = \bigcup_{n=1}^\infty X_n for some X_n \in \mathcal{S} with \mu(X_n) < \infty), the restriction of this measure to σ(𝒮) is the unique extension of μ to σ(𝒮).^[2]^[3]^[1]

Proof outline

The proof of Carathéodory's extension theorem unfolds through a sequence of constructions and verifications that extend a premeasure on a semi-ring to a measure on the generated σ-algebra. The initial step constructs an outer measure \mu^* on the power set of the space X by defining, for any subset E \subset X,

\mu^*(E) = \inf\left\{ \sum_{n=1}^\infty \mu(S_n) : S_n \in \mathcal{S}, E \subset \bigcup_{n=1}^\infty S_n \right\},

where \mathcal{S} is the semi-ring and the infimum is taken over all countable covers of E by sets in \mathcal{S}, with \mu(\emptyset) = 0. This \mu^* satisfies the properties of an outer measure, including monotonicity and countable subadditivity, and extends \mu on \mathcal{S} in the sense that \mu^*(S) = \mu(S) for all S \in \mathcal{S}.^[3]^[18] Next, the Carathéodory measurable sets are defined as the collection \mathcal{M} of subsets E \subset X such that for every A \subset X,

\mu^*(A) = \mu^*(A \cap E) + \mu^*(A \cap E^c).

This condition ensures that E "splits" any set A additively with respect to \mu^*.^[3]^[18] The collection \mathcal{M} is then verified to form a σ-algebra containing \mathcal{S}, as it includes the empty set and X, is closed under complements, and is closed under countable unions. Moreover, \mu^* restricted to \mathcal{M} yields a measure, with countable additivity holding for disjoint unions in \mathcal{M}; this relies on σ-finiteness of the space to control approximations in the outer measure construction.^[3]^[18] Finally, the σ-algebra \sigma(\mathcal{S}) generated by \mathcal{S} is shown to be contained in \mathcal{M}, so \mu^* provides a measure on \sigma(\mathcal{S}) extending \mu. Under the assumption of σ-finiteness (where X is a countable union of sets of finite \mu-measure), this extension is unique among measures on \sigma(\mathcal{S}).^[3]^[18]

Uniqueness considerations

Conditions for uniqueness

In Carathéodory's extension theorem, the uniqueness of the measure extension from a premeasure \mu defined on a semi-ring S to the \sigma-algebra \sigma(S) holds under the condition that \mu is \sigma-finite.^[19] A premeasure \mu is \sigma-finite if the underlying space can be covered by a countable collection of sets in S each of finite \mu-measure./02:_Probability_Spaces/2.08:_Existence_and_Uniqueness) In this case, the Carathéodory extension is the unique measure on \sigma(S) that agrees with \mu on S.^[20] The proof of uniqueness relies on the \pi-\lambda theorem, which states that if two measures agree on a \pi-system generating the \sigma-algebra, then they agree on the entire \sigma-algebra.^[21] Here, the semi-ring S serves as a \pi-system, and the \pi-\lambda theorem (or equivalently, the monotone class theorem) ensures that any two extensions agreeing on S must coincide on the algebra generated by S, and thus on \sigma(S).^[19] This argument requires \sigma-finiteness to handle the approximation of sets by countable covers and to control the measures on finite-measure subsets.^[21] When \mu is not \sigma-finite, the extension to \sigma(S) need not be unique, as multiple measures may agree on S but differ on \sigma(S).^[20] Such non-uniqueness arises because the outer measure construction may allow for different completions or extensions beyond the Carathéodory measurable sets./02:_Probability_Spaces/2.08:_Existence_and_Uniqueness) A special case occurs when \mu is finite, meaning \mu(X) < \infty for the whole space X. Finite premeasures are inherently \sigma-finite, so uniqueness holds without invoking the full \sigma-finiteness condition separately.^[19] This ensures that the extension is unique even in settings where broader \sigma-finiteness might otherwise be needed.^[20]

Counting measure example

Let X = \mathbb{R}, an uncountable set, and let S be the algebra generated by all half-open intervals [a, b) with a, b \in \mathbb{Q}. Define the premeasure \mu on S by \mu(\emptyset) = 0 and \mu(E) = \infty for every non-empty E \in S.^[22] This premeasure is not \sigma-finite, as any cover of \mathbb{R} by sets from S requires infinitely many non-empty sets, each with measure \infty. The outer measure \mu^* induced by \mu is given by

\mu^*(A) = \inf\left\{ \sum_{n=1}^\infty \mu(E_n) \;\middle|\; A \subseteq \bigcup_{n=1}^\infty E_n, \; E_n \in S \right\}

for A \subseteq \mathbb{R}. For any non-empty A, any countable cover by sets from S must include at least one non-empty interval, so the sum is \infty, yielding \mu^*(A) = \infty. The empty set has measure 0. The Carathéodory measurable sets are all subsets of \mathbb{R}, and the extension \mu restricts to \mu(E) = 0 if E = \emptyset and \infty otherwise.^[22] A different extension to the Borel \sigma-algebra \sigma(S) (generated by S) is the counting measure \nu(E) = |E| if E is finite and \infty otherwise (with \infty for countable or uncountable infinite sets). Both \mu and \nu agree on S, since every non-empty set in S is infinite (uncountable) and thus has measure \infty under \nu. However, they differ on \sigma(S); for example, on a singleton \{x\}, which is Borel, \mu(\{x\}) = \infty while \nu(\{x\}) = 1. This demonstrates non-uniqueness due to the lack of \sigma-finiteness.^[22]

Rationals example

Consider the set X = \mathbb{Q} \cap [0,1), the rational numbers in the half-open unit interval. Define the collection \Sigma_0 as the algebra consisting of all finite disjoint unions of sets of the form (a, b] \cap X, where a, b \in \mathbb{Q} and $0 \leq a < b < 1. Each non-empty set in \Sigma_0 is countably infinite, as the rationals are dense. Define the set function \mu_0: \Sigma_0 \to [0, \infty] by \mu_0(\emptyset) = 0 and \mu_0(E) = \infty for every non-empty E \in \Sigma_0. This \mu_0 is a premeasure on \Sigma_0, as it is finitely (and even countably) additive, since the only disjoint unions in \Sigma_0 either yield the empty set or a non-empty set with measure \infty. The \sigma-algebra \Sigma generated by \Sigma_0 is the power set of X, because singletons \{x\} for x \in X belong to \Sigma; specifically, \{x\} = \bigcap_{n=1}^\infty (x - 1/n, x] \cap X. Carathéodory's extension theorem guarantees the existence of measures on \Sigma extending \mu_0, but uniqueness fails because \mu_0 is not \sigma-finite: X cannot be covered by countably many sets from \Sigma_0 of finite \mu_0-measure, as all non-empty such sets have measure \infty. Explicitly, the counting measure \mu_1(E) = |E| (cardinality, with \infty for infinite E) extends \mu_0, since every non-empty E \in \Sigma_0 is infinite, so \mu_1(E) = \infty. Another extension is \mu_2(E) = 2 |E| (with the convention $2 \cdot \infty = \infty), which also satisfies \mu_2(E) = \infty for E \in \Sigma_0. Both \mu_1 and \mu_2 are \sigma-additive on \Sigma and \sigma-finite (as X is countable, covered by singletons of finite measure). However, they differ on finite subsets: for a finite A \subset X with |A| = n < \infty, \mu_1(A) = n while \mu_2(A) = 2n. They also differ on X itself, both assigning \infty, but on subsets like \mathbb{Q} \cap [0, 1/2), the measures coincide at \infty yet vary in their distribution over finite subcollections of rationals. This demonstrates how non-\sigma-finiteness of the premeasure permits multiple extensions, even to a \sigma-finite measure space.

Fubini theorem example

Consider the space X = [0,1] \times [0,1], equipped with the semi-ring \mathcal{S} consisting of finite unions of rectangles of the form I \times F, where I is an interval with rational endpoints in [0,1] and F is a finite set of rational points in [0,1]. Define the premeasure \mu on \mathcal{S} by \mu(I \times F) = m(I) \cdot \#(F), where m denotes the Lebesgue measure on [0,1] and \# denotes the counting measure.^[23]^[1] By Carathéodory's extension theorem, this premeasure extends to measures on the \sigma-algebra generated by \mathcal{S}, which includes the product \sigma-algebra \mathcal{L}([0,1]) \times \mathcal{P}([0,1]). However, due to the lack of \sigma-finiteness of the counting measure on the uncountable set [0,1], the extension is not unique. Two distinct extensions \mu_1 and \mu_2 agree on \mathcal{S} but differ elsewhere. Specifically, \mu_1(E) = \int_{[0,1]} \#(E_x) \, dm(x) for measurable E, where E_x = \{y \in [0,1] : (x,y) \in E\}, while \mu_2(E) = \int_{[0,1]} m(E^y) \, d\#(y), where E^y = \{x \in [0,1] : (x,y) \in E\}.^[23]^[1] This non-uniqueness manifests in the treatment of the diagonal set D = \{(x,x) : x \in [0,1]\}. Under \mu_1, \mu_1(D) = \int_{[0,1]} \#(\{x\}) \, dm(x) = \int_{[0,1]} 1 \, dm(x) = 1, whereas under \mu_2, \mu_2(D) = \int_{[0,1]} m(\{y\}) \, d\#(y) = \int_{[0,1]} 0 \, d\#(y) = 0. Both extensions agree on \mathcal{S}, as rectangles I \times F have slices yielding the premeasure value in either construction.^[23]^[1] The connection to Fubini's theorem arises from its reliance on \sigma-finiteness and completeness for interchanging iterated integrals over product measures. Here, without \sigma-finiteness, the differing extensions lead to disparate iterated integrals for the characteristic function f = 1_D: \int_{[0,1]} \left( \int_{[0,1]} f(x,y) \, d\#(y) \right) dm(x) = 1 under the first ordering, but \int_{[0,1]} \left( \int_{[0,1]} f(x,y) \, dm(x) \right) d\#(y) = 0 under the second, illustrating the failure without these assumptions.^[23]^[1]

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