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Set function

A set function is a mathematical function whose domain consists of a collection of sets, typically subsets of a given , and which assigns to each such set an element from a , often the extended real numbers [-\infty, \infty]. In the context of , set functions frequently map to non-negative extended reals [0, \infty] and exhibit properties such as monotonicity (if A \subseteq B, then \mu(A) \leq \mu(B)) and (\mu(A \cup B) \leq \mu(A) + \mu(B)). Set functions form the foundation of measure theory, where a measure is defined as a countably additive set function \mu on a \sigma-algebra of sets, satisfying \mu(\emptyset) = 0 and \mu\left(\bigcup_{n=1}^\infty A_n\right) = \sum_{n=1}^\infty \mu(A_n) for disjoint sets A_n. This additivity property distinguishes measures from more general set functions, enabling the rigorous definition of integrals and probabilities. Notable examples include the Lebesgue measure on \mathbb{R}^n, which assigns volumes to measurable sets and extends the notion of length, area, and volume; the Dirac measure \delta_x, defined by \delta_x(A) = 1 if x \in A and 0 otherwise, concentrating mass at a point; and the counting measure, which counts the cardinality of finite sets and assigns infinity to infinite ones. Beyond measure theory, set functions appear in probability as probability measures, which are measures normalized so that \mu(X) = 1 for the entire space X, and in , where they model quantities like the inclusion-exclusion principle for unions of sets. Properties such as continuity from below or above—where \mu\left(\bigcup A_n\right) = \lim \mu(A_n) for increasing or decreasing sequences of sets—further characterize useful set functions in and approximation theorems. These concepts underpin advanced topics like outer measures, for constructing measures from premeasures, and applications in and stochastic processes.

Definitions and Basic Concepts

Formal Definition

In , a set function is a \mu whose is a collection of subsets of a given set X, typically the power set \mathcal{P}(X), and whose is the extended s \overline{\mathbb{R}} = \mathbb{R} \cup \{ -\infty, +\infty \}, thereby assigning a real number or to each such . Formally, \mu: \mathcal{P}(X) \to \overline{\mathbb{R}}, though the is often restricted to a subcollection like an or \sigma- of subsets to facilitate specific properties. The standard notation is \mu(A) for any A \subseteq X. The concept of set functions arose in the early 20th century as part of the foundational developments in measure theory, pioneered by mathematicians such as and . Lebesgue introduced key ideas in his 1902 work on and the measure of sets, while Carathéodory provided a rigorous axiomatic framework in 1914 that generalized measure constructions. A simple example of a 0-1 valued set function is the \delta_x at a fixed point x \in X, defined by \delta_x(A) = \begin{cases} 1 & \text{if } x \in A, \\ 0 & \text{if } x \notin A, \end{cases} for any A \subseteq X, which serves as the of the \{x\} in this context.

Elementary Properties

One fundamental property often imposed on set functions μ: 2^X → ℝ is the at the , where μ(∅) = 0. This condition ensures that the empty subset carries no inherent value or measure, serving as a foundational in definitions of capacities and fuzzy measures. Although not universally required for arbitrary set functions, it is a standard assumption that facilitates consistency in applications such as and . The value assigned to the full set X, denoted μ(X), typically represents the total or extent of the universe under consideration. In normalized settings, such as probabilistic or fuzzy measure contexts, μ(X) is set to 1 to reflect , while in more general frameworks it may take any non-negative real value or , indicating the overall scale. Non-negativity is another prevalent elementary , requiring μ(A) ≥ 0 for all A ⊆ X. This ensures that set functions model positive quantities like sizes or beliefs without allowing negative assignments, and it is explicitly part of definitions for capacities and fuzzy measures. Set functions are also distinguished by their range: finite-valued ones satisfy μ(A) < ∞ for every A ⊆ X, common in bounded domains, whereas infinite-valued functions permit μ(A) = ∞, as seen in extended real-valued measures for unbounded spaces. In contexts involving monotonic set functions, a basic inequality holds: for any subsets A, B ⊆ X, μ(A ∪ B) ≥ \max(μ(A), μ(B)). This derives from the increasing nature of such functions and provides a minimal bound on unions without assuming additivity.

Classifications and Types

Additive and Subadditive Functions

A set function \mu: \mathcal{A} \to [0, \infty) defined on a collection \mathcal{A} of subsets of a set X is finitely additive if \mu(\emptyset) = 0 and, for any finite collection of pairwise disjoint sets \{A_1, \dots, A_n\} \subseteq \mathcal{A} such that \bigcup_{k=1}^n A_k \in \mathcal{A}, \mu\left(\bigcup_{k=1}^n A_k\right) = \sum_{k=1}^n \mu(A_k). In particular, for two disjoint sets A, B \in \mathcal{A}, finite additivity yields \mu(A \cup B) = \mu(A) + \mu(B). Countable additivity, or \sigma-additivity, extends this property to infinite collections: \mu is \sigma-additive if \mu(\emptyset) = 0 and, for any countable collection of pairwise \{A_n\}_{n=1}^\infty \subseteq \mathcal{A} such that \bigcup_{n=1}^\infty A_n \in \mathcal{A}, \mu\left(\bigcup_{n=1}^\infty A_n\right) = \sum_{n=1}^\infty \mu(A_n). Every \sigma-additive set function is finitely additive, as one can take all but finitely many A_n to be empty. However, the converse does not hold; there exist finitely additive set functions that fail to be \sigma-additive. For instance, a Banach provides an example of a finitely additive on the power set of the natural numbers that extends the notion of asymptotic but is not \sigma-additive, as it assigns positive measure to certain unbounded sets whose countable disjoint union would violate countable additivity. A set function \mu is subadditive if, for any finite or countable collection \{A_i\}_{i \in I} \subseteq \mathcal{A} (with I finite or countable) such that \bigcup_{i \in I} A_i \in \mathcal{A}, \mu\left(\bigcup_{i \in I} A_i\right) \leq \sum_{i \in I} \mu(A_i). In particular, for two sets A, B \in \mathcal{A}, subadditivity implies \mu(A \cup B) \leq \mu(A) + \mu(B). The Lebesgue outer measure exemplifies a \sigma-subadditive set function. A set function \mu is superadditive if, for any finite or countable collection of pairwise \{A_i\}_{i \in I} \subseteq \mathcal{A} (with I finite or countable) such that \bigcup_{i \in I} A_i \in \mathcal{A}, \mu\left(\bigcup_{i \in I} A_i\right) \geq \sum_{i \in I} \mu(A_i). For two A, B \in \mathcal{A}, this reduces to \mu(A \cup B) \geq \mu(A) + \mu(B). Subadditivity implies bounds involving overlaps when combined with non-negativity. Specifically, for any A, B \in \mathcal{A}, \mu(A \cup B) + \mu(A \cap B) \leq \mu(A) + \mu(B). To sketch the proof, partition A = (A \cap B) \cup (A \setminus B) and B = (A \cap B) \cup (B \setminus A). By subadditivity (applied to two sets), \mu(A) \leq \mu(A \cap B) + \mu(A \setminus B), \quad \mu(B) \leq \mu(A \cap B) + \mu(B \setminus A). Adding these yields \mu(A) + \mu(B) \geq 2\mu(A \cap B) + \mu(A \setminus B) + \mu(B \setminus A). Now apply subadditivity to the three pairwise disjoint sets A \setminus B, B \setminus A, and A \cap B, whose union is A \cup B: \mu(A \cup B) \leq \mu(A \setminus B) + \mu(B \setminus A) + \mu(A \cap B). Adding \mu(A \cap B) to both sides gives \mu(A \cup B) + \mu(A \cap B) \leq \mu(A \setminus B) + \mu(B \setminus A) + 2\mu(A \cap B) \leq \mu(A) + \mu(B), as required. This inequality highlights how subadditivity controls the "overlap penalty" in unions, bounding the joint measure by the individual measures.

Measures and Capacities

In measure theory, a measure is defined as a non-negative set function \mu: \mathcal{A} \to [0, \infty] defined on a \sigma-algebra \mathcal{A} over a set X, satisfying \sigma-additivity: for any countable collection of pairwise disjoint sets \{A_n\}_{n=1}^\infty \subseteq \mathcal{A} with union in \mathcal{A}, \mu\left(\bigcup_{n=1}^\infty A_n\right) = \sum_{n=1}^\infty \mu(A_n), and normalized by \mu(\emptyset) = 0. This extends finite additivity to countable unions, ensuring consistency in handling infinite decompositions on the domain restricted to a \sigma-algebra. Signed measures generalize positive measures by allowing negative values, defined as set functions \nu: \mathcal{A} \to [-\infty, \infty) on a \sigma-algebra \mathcal{A} that are countably additive and satisfy \nu(\emptyset) = 0, with the additional condition that \nu does not take both +\infty and -\infty. The total variation of a signed measure \nu on a set A \in \mathcal{A} is given by |\nu|(A) = \sup \left\{ \sum_{i=1}^n |\nu(E_i)| : \{E_i\}_{i=1}^n \subseteq \mathcal{A} \text{ disjoint, } \bigcup_{i=1}^n E_i \subseteq A \right\}, where the supremum is over all finite partitions; this yields a positive finite measure |\nu| that bounds the oscillation of \nu. By the Jordan decomposition theorem, any signed measure decomposes uniquely into \nu = \nu^+ - \nu^-, where \nu^+ and \nu^- are positive measures with disjoint supports, and |\nu| = \nu^+ + \nu^-. Capacities represent a broader class of monotone set functions \gamma: 2^X \to [0,1] on the power set of X, satisfying \gamma(\emptyset) = 0, \gamma(X) = 1, and monotonicity: if A \subseteq B \subseteq X, then \gamma(A) \leq \gamma(B). Outer capacities approximate sets from above using open covers, while inner capacities approximate from below using closed subsets; these are foundational in non-additive integration, such as the Choquet integral. Choquet capacities, introduced as a specific type, are monotone functions that may also satisfy alternating properties—for sets A, B \subseteq X, \gamma(A \cup B) + \gamma(A \cap B) \leq \gamma(A) + \gamma(B)—enabling the representation of non-linear expectations in decision theory and potential theory. In topological spaces, measures often exhibit regularity properties that align their values with the topology. A measure \mu on the Borel \sigma-algebra is outer regular if for every Borel set A, \mu(A) = \inf \{ \mu(U) : U \text{ open}, A \subseteq U \}. Inner regularity requires \mu(A) = \sup \{ \mu(K) : K \text{ compact}, K \subseteq A \}, and full regularity combines both; these ensure measures are determined by their values on compact or open sets, facilitating approximation in locally compact Hausdorff spaces. Completeness further characterizes measures: a measure space (X, \mathcal{A}, \mu) is complete if every subset of a null set (i.e., set of \mu-measure zero) is measurable and has measure zero, often achieved by extending the \sigma-algebra to include all such subsets.

Advanced Properties

Monotonicity and Continuity Conditions

Monotonicity is a fundamental property of many set functions, particularly those that are non-negative. A set function \mu: \mathcal{A} \to [0, \infty], where \mathcal{A} is a collection of subsets of a set X containing the , is said to be if for all A, B \in \mathcal{A} with A \subseteq B, it holds that \mu(A) \leq \mu(B). This property ensures that the function respects the partial order of set inclusion, making it suitable for applications in optimization and where larger sets should not receive smaller values. Beyond monotonicity, continuity conditions provide finer control over the behavior of set functions under limits of sequences of sets. Continuity from below requires that for any increasing sequence of sets \{A_n\}_{n=1}^\infty \subseteq \mathcal{A} with A_n \subseteq A_{n+1} for all n and \bigcup_{n=1}^\infty A_n = A \in \mathcal{A}, the satisfies \lim_{n \to \infty} \mu(A_n) = \mu(A). Similarly, continuity from above stipulates that for any decreasing sequence \{A_n\}_{n=1}^\infty \subseteq \mathcal{A} with A_n \supseteq A_{n+1} for all n, \bigcap_{n=1}^\infty A_n = A \in \mathcal{A}, and \mu(A_1) < \infty, it holds that \lim_{n \to \infty} \mu(A_n) = \mu(A). These conditions generalize the intuitive notion that measures should behave continuously with respect to nested unions and intersections, though they apply to general monotone set functions without requiring additivity. A key relationship exists between \sigma-additivity and continuity from below. Suppose \mu is \sigma-additive, meaning that for any countable collection of pairwise \{B_n\}_{n=1}^\infty \subseteq \mathcal{A} with \bigcup_{n=1}^\infty B_n \in \mathcal{A}, \mu\left(\bigcup_{n=1}^\infty B_n\right) = \sum_{n=1}^\infty \mu(B_n). To show continuity from below, consider an increasing A_n \uparrow A. Define disjoint sets B_1 = A_1 and B_n = A_n \setminus A_{n-1} for n \geq 2. Then A = \bigcup_{n=1}^\infty B_n, so by \sigma-additivity, \mu(A) = \sum_{n=1}^\infty \mu(B_n). The partial sums are \sum_{k=1}^n \mu(B_k) = \mu(A_n) by finite additivity (which follows from \sigma-additivity), and since the terms \mu(B_n) \geq 0, the for series yields \lim_{n \to \infty} \mu(A_n) = \sum_{n=1}^\infty \mu(B_n) = \mu(A). Not all monotone set functions satisfy these continuity conditions, leading to discontinuities at certain limits. For instance, consider the set function \mu on the power set of \mathbb{N} defined by \mu(A) = 0 if A is finite and \mu(A) = 1 if A is infinite. This \mu is monotone, as finite subsets of infinite sets have measure 0 while infinite subsets have measure 1, and finite subsets of finite sets preserve 0. However, it fails continuity from below: the increasing sequence A_n = \{1, 2, \dots, n\} satisfies \mu(A_n) = 0 for all n, but \bigcup_n A_n = \mathbb{N} has \mu(\mathbb{N}) = 1 \neq \lim_n \mu(A_n). Similarly, Dirac-like set functions, such as the point mass \delta_p(A) = 1 if p \in A and 0 otherwise for a fixed point p, exhibit jumps precisely at sets containing or excluding p, though they satisfy continuity under the standard sequence conditions when \sigma-additive.

Inner and Outer Approximations

In measure theory, outer measures provide a way to approximate the size of arbitrary sets from above using coverings from a predefined equipped with a set function. Given a set function μ defined on a collection of sets 𝒞 (often an or ), the μ* induced by μ on the power set of the ambient space X is defined as \mu^*(A) = \inf \left\{ \sum_{i=1}^\infty \mu(E_i) : A \subseteq \bigcup_{i=1}^\infty E_i, \, E_i \in \mathcal{C} \right\} for any subset A ⊆ X, where the infimum is taken over all countable covers of A by sets from 𝒞, and the sum is understood to be infinite if no such cover exists with finite total measure. This construction extends μ to all subsets while preserving an upper bound on "size." Outer measures satisfy monotonicity, meaning if A ⊆ B then μ*(A) ≤ μ*(B), and subadditivity, so μ*(∪{i=1}^∞ A_i) ≤ ∑{i=1}^∞ μ*(A_i) for any countable collection of sets {A_i}. These properties ensure that outer measures are non-negative, with μ*(∅) = 0, and finitely subadditive as well. Dually, inner measures approximate sets from below using contained subsets from a suitable family, often closed or compact sets in a topological context. The inner measure μ_* induced by μ is given by \mu_*(A) = \sup \left\{ \mu(K) : K \subseteq A, \, K \text{ compact (or closed)} \right\} for A ⊆ X, where the supremum is over all compact (or closed) subsets K of A on which μ is defined. Inner measures are superadditive: if A and B are disjoint, then μ_(A ∪ B) ≥ μ_(A) + μ_(B), and they satisfy μ_(A) ≤ μ_(B) if A ⊆ B, along with μ_(X) = μ(X) if μ is defined on the whole space. This duality allows inner measures to capture a lower bound on the "size" of A, complementing the outer approximation. A key tool for identifying measurable sets within this framework is the Carathéodory criterion, which characterizes sets that behave additively with respect to the . A set E ⊆ X is Carathéodory measurable if for all test sets T ⊆ X, \mu^*(T) = \mu^*(T \cap E) + \mu^*(T \cap E^c). This condition ensures that E splits any test set T into measurable pieces without altering the total . The collection of Carathéodory measurable sets forms a on which the restriction of μ* is a complete measure. For a set A, if the inner measure μ_(A) equals the outer measure μ(A), then A is measurable in the sense that it coincides with the measure on the generated by the approximations, ensuring uniqueness in the extension process. This agreement provides a regularity that aligns the , facilitating the construction of measures on larger σ-algebras.

Relationships and Equivalences

Connections to Measures

In measure theory, general set functions such as provide a framework for identifying measurable sets via : a E of the ambient X is measurable if for every set A \subseteq X, m^*(A) = m^*(A \cap E) + m^*(A \setminus E), where m^* is the . For such measurable sets, the inner measure m_*(E), defined as \sup \{ m(F) \mid F \text{ measurable}, F \subseteq E \} or equivalently (when applicable) as m(A) - m^*(A \setminus E) for a measurable cover A \supseteq E, equals the outer measure m^*(E). This ensures that measurable sets capture a precise notion of "size" without ambiguity between approximation from below and above. The collection \Sigma of all such measurable sets forms a \sigma-algebra, and restricting the outer measure to \Sigma yields a countably additive measure \mu: \Sigma \to [0, \infty], thereby defining a measure space (X, \Sigma, \mu). This construction, known as the Carathéodory extension, transforms a general subadditive set function (like an ) into a complete measure on the \sigma-algebra of measurable sets, bridging arbitrary set functions to the rigorous structure required for integration and analysis. Within measure spaces, two measures \mu and \nu on the same \sigma-algebra are equivalent if they share the same null sets, meaning \mu(E) = 0 if and only if \nu(E) = 0 for all measurable E, allowing them to agree almost everywhere with respect to each other. This equivalence relation preserves the essential properties of the measures while ignoring differences on sets of measure zero, facilitating comparisons in probability and analysis. A key connection arises through : if \mu \ll \nu (i.e., \nu(E) = 0 implies \mu(E) = 0), the Radon-Nikodym theorem guarantees the existence of a f \geq 0, called the Radon-Nikodym derivative \frac{d\mu}{d\nu}, such that for every measurable set E, \mu(E) = \int_E \frac{d\mu}{d\nu} \, d\nu. This derivative represents \mu as an with respect to \nu, enabling the decomposition of measures and the study of densities in set function contexts. Furthermore, the extends to integrals defined over measure spaces derived from set functions, where if a of measurable functions f_n converges to f and |f_n| \leq g with g , then \int f_n \, d\mu \to \int f \, d\mu. This result justifies limit interchanges in approximations by simple functions supported on measurable sets, linking set functions to robust calculus.

Topological and Algebraic Links

In topological spaces, set functions often manifest as Borel measures, which are defined on the Borel generated by the open sets of the . The Borel consists of the smallest containing all open sets, ensuring that Borel measures assign values to sets derived from the topological structure while maintaining σ-additivity and non-negativity. This integration allows set functions to respect the topological properties, such as openness and closedness, facilitating the study of continuity and approximation in spaces like or Hausdorff topologies. A key topological property of such measures is regularity, which quantifies how well measurable sets can be approximated by open or compact subsets. A μ is outer regular if for every E, μ(E) equals the infimum of μ(U) over all open sets U containing E, allowing approximation from above by opens. Similarly, μ is inner regular if μ(E) = sup {μ(K) : K compact, K ⊆ E}, providing approximation from below by compact sets. In locally compact Hausdorff spaces, many s, such as those finite on compacts, exhibit both inner and outer regularity, enhancing their utility in and theorems. Algebraically, set functions link to group structures through invariance under group actions, particularly in locally compact groups where Haar measures serve as canonical examples. A left Haar measure on a locally compact Hausdorff group G is a non-zero regular Borel measure μ that is left-invariant, meaning μ(xE) = μ(E) for all x in G and Borel sets E, with finite measure on compact sets. This invariance preserves the algebraic symmetry of the group, enabling the definition of integrals over group elements and applications in representation theory. Right Haar measures are defined analogously, and in compact or abelian groups, left and right versions coincide up to scalar multiples. In non-Hausdorff topological s, regularity of measures requires careful consideration, as the lack of separation axioms can complicate and . While inner regularity is typically defined using compact subsets, some formulations adapt to compact closed sets to ensure well-behaved approximations, though standard literature often retains compact sets for consistency. Additional conditions, such as the space being completely or the measure being tight, may be imposed to guarantee properties like inner regularity on open sets.

Examples

Lebesgue Measure

The serves as a fundamental example of a set function on the \mathbb{R}^n, providing a rigorous of , area, and to arbitrary measurable subsets. Its construction begins with the definition of the \mu^*, which assigns to any subset A \subseteq \mathbb{R}^n the infimum of the sums of volumes of countable coverings by open s: \mu^*(A) = \inf \left\{ \sum_k v(R_k) : A \subseteq \bigcup_k R_k, \, R_k \text{ open rectangles} \right\}, where v(R) denotes the of the R. This outer measure is then restricted to the \sigma- of Lebesgue measurable sets via the Carathéodory criterion, yielding the complete Lebesgue measure \lambda on \mathbb{R}^n. For a half-open [a_1, b_1) \times \cdots \times [a_n, b_n) in \mathbb{R}^n, the measure is given by \lambda([a_1, b_1) \times \cdots \times [a_n, b_n)) = \prod_{i=1}^n (b_i - a_i), which extends additively to finite disjoint unions of such s and further via the Carathéodory extension to the full \sigma-. The exhibits key properties that make it a prototypical measure-theoretic set function. It is translation-invariant, meaning \lambda(A + t) = \lambda(A) for any measurable A \subseteq \mathbb{R}^n and t \in \mathbb{R}^n. Additionally, it is \sigma-finite, as \mathbb{R}^n can be covered by countably many sets of finite measure, such as balls of radius 1. On the Lebesgue \sigma-algebra, which includes all Borel sets and their completions with , the measure is complete: any subset of a null set is also measurable with measure zero. In infinite-dimensional settings, such as separable Hilbert spaces, no direct analog of the exists due to the absence of a locally finite, translation-invariant measure on the entire space. Instead, Gaussian measures provide a natural extension, defined via cylindrical approximations and characterized by their mean and covariance operator; these measures are quasi-invariant under translations but not translation-invariant, assigning positive measure to neighborhoods while having full support. The on \mathbb{R}^n is unique up to sets among all \sigma-finite, translation-invariant measures on the Borel \sigma-algebra, as established by its role as the on the additive group \mathbb{R}^n, normalized such that \lambda([0,1)^n) = 1.

Dirac Measure

The Dirac measure \delta_x, for a fixed point x in a X, is a basic example of a set function defined by \delta_x(A) = 1 if x \in A and $0 otherwise, for subsets A \subseteq X. It is a probability measure on the power set or Borel \sigma-algebra, concentrating all mass at x. This set function is finitely additive (and \sigma-additive) and serves as a Dirac delta distribution in integration, with key properties including translation-invariance in the sense that \delta_{x+y}(A) = \delta_x(A - y). It is widely used in probability to model point masses and in physics for impulses.

Counting Measure

The counting measure on a set X assigns to each A \subseteq X the |A| if A is finite, and \infty otherwise. It is a set function on the power set, \sigma-additive, and monotone, but not \sigma-finite unless X is countable. On uncountable sets like \mathbb{R}, it extends the notion of "size" beyond finite measures, appearing in and as the on discrete groups.

Invariant Set Functions

Invariant set functions, particularly those that are translation-invariant, maintain their value under shifts by elements of an underlying group. A set function \mu defined on subsets of a group G is translation-invariant if \mu(A + x) = \mu(A) for all measurable sets A \subseteq G and all x \in G, where A + x = \{a + x \mid a \in A\}. This property is central in finitely additive cases, where \mu satisfies additivity over disjoint finite unions but not necessarily countable ones, allowing extensions beyond standard measures. A prominent example is the Banach measure on \mathbb{R}, which is a finitely additive, translation-invariant set function extending the from the Lebesgue \sigma-algebra to the power set of \mathbb{R}, normalized such that \mu([0,1]) = 1. Originally constructed by in 1923 using group-theoretic methods, modern proofs employ the Hahn-Banach theorem; this measure agrees with on measurable sets but is not \sigma-additive, enabling it to assign measures to non-measurable sets while preserving invariance under translations. For additive invariant functions on \mathbb{R}^n, setting \mu([0,1)^n) = 1 ensures consistency with volume properties, though such extensions are not unique. In the context of locally compact groups, Haar measure provides a canonical example of an invariant set function that is \sigma-additive. A left Haar measure \mu on a G satisfies \mu(gA) = \mu(A) for all compactly supported measurable sets A and g \in G, and is unique up to positive scalar multiples; right Haar measures satisfy the analogous right-invariance. The existence of such measures follows from the applied to positive linear functionals on the space of continuous compactly supported functions C_c(G), yielding a regular Borel measure that is invariant.

Extensions and Constructions

From Semirings to Algebras

A semiring of sets is a non-empty collection \mathcal{P} of subsets of a set X that is closed under finite intersections, meaning that if E, F \in \mathcal{P}, then E \cap F \in \mathcal{P}, and such that for any E, F \in \mathcal{P} with E \subseteq F, the difference F \setminus E can be expressed as a finite disjoint union of sets from \mathcal{P}. This structure provides a foundational class for defining set functions, often starting with finitely additive functions on \mathcal{P}. Semirings are particularly useful in measure theory because they allow systematic extensions while preserving key properties like monotonicity. To extend a set function \mu: \mathcal{P} \to [0, \infty] defined on a \mathcal{P} to the \mathcal{A} generated by \mathcal{P}, one defines \mu on \mathcal{A} using finite s of elements from \mathcal{P}. Specifically, every set A \in \mathcal{A} can be written as a finite A = \bigsqcup_{i=1}^n R_i where each R_i \in \mathcal{P}, and then \mu(A) = \sum_{i=1}^n \mu(R_i). This definition ensures that if \mu is finitely additive and on \mathcal{P}, then the extended \mu inherits these properties on \mathcal{A}, maintaining finite additivity for in the algebra and monotonicity for nested sets. The extension is provided that the whole X can be covered by a finite of sets from \mathcal{P}, ensuring that the \mathcal{A} includes X and that no alternative representations affect the values of \mu. This finite covering condition guarantees that the measure remains well-defined and consistent across equivalent decompositions in \mathcal{A}.

Carathéodory Extension

The Carathéodory extension theorem provides a method to extend a countably additive set function, or pre-measure, defined on a ring of subsets to a complete measure on the σ-algebra it generates. Consider a set X and a ring \mathcal{R} of subsets of X, equipped with a pre-measure \mu: \mathcal{R} \to [0, \infty] that is countably additive, meaning \mu\left(\bigcup_{n=1}^\infty E_n\right) = \sum_{n=1}^\infty \mu(E_n) for any countable collection of pairwise disjoint sets E_n \in \mathcal{R} whose union lies in \mathcal{R}. This setup allows construction of an outer measure on the power set \mathcal{P}(X), which serves as the foundation for the extension. The \mu^* is defined for any A \subset X by \mu^*(A) = \inf\left\{ \sum_{n=1}^\infty \mu(E_n) : E_n \in \mathcal{R}, \, A \subset \bigcup_{n=1}^\infty E_n \right\}, with the convention that the infimum over the empty collection is \infty. This \mu^* inherits properties such as monotonicity and countable from \mu, and it extends \mu in the sense that \mu^*(E) = \mu(E) for all E \in \mathcal{R}. A E \subset X is Carathéodory measurable if it satisfies the : for every A \subset X, \mu^*(A) = \mu^*(A \cap E) + \mu^*(A \cap E^c), where E^c = X \setminus E. The collection \mathcal{M} of all Carathéodory measurable sets forms a containing \mathcal{R}, and the restriction \mu|_{\mathcal{M}} is a complete measure that extends \mu, meaning \mu(E) = \mu^*(E) for E \in \mathcal{R} and \mu is countably additive on \mathcal{M}. This extension theorem, originally formulated by , guarantees that \mathcal{M} includes the generated by \mathcal{R}. If \mu is σ-finite, meaning X can be covered by countably many sets of finite \mu-measure, then the extension is unique: any other measure on the σ-algebra generated by \mathcal{R} that agrees with \mu on \mathcal{R} must coincide with \mu|_{\mathcal{M}}. Without σ-finiteness, uniqueness may fail, as multiple extensions can exist. The proof begins by verifying that \mathcal{M} is a σ-algebra, showing closure under complements (directly from the splitting condition) and countable unions (using subadditivity and the splitting property iteratively). Countable additivity of \mu|_{\mathcal{M}} follows from the outer measure's subadditivity and the measurability condition, which prevents "overlaps" in approximations. Completeness arises because σ-additivity of the original \mu on \mathcal{R} ensures that null sets (sets with \mu^* = 0) and their subsets are measurable, with measure zero, making the extension complete.

Outer Measure Restrictions

In measure theory, the restriction of an outer measure refers to the process of limiting its domain to the σ-algebra of Carathéodory-measurable sets, thereby yielding a complete measure. An outer measure μ* on a set X is a function from the power set P(X) to [0, ∞] satisfying μ*(∅) = 0, monotonicity (if A ⊆ B, then μ*(A) ≤ μ*(B)), and countable subadditivity (μ*(∪{n=1}^∞ A_n) ≤ ∑{n=1}^∞ μ*(A_n) for any countable collection {A_n} ⊆ P(X)). Unlike a full measure, an outer measure may not be countably additive on all subsets, but its restriction addresses this limitation. A A ⊆ X is μ*-measurable (or Carathéodory-measurable) if, for every Y ⊆ X, μ*(Y) = μ*(Y ∩ A) + μ*(Y \ A). This condition, known as , ensures that A "splits" any test set Y additively with respect to the . Carathéodory's theorem states that the collection M of all μ*-measurable sets forms a on X, and the restriction μ*|_M : M → [0, ∞], defined by μ(A) = μ*(A) for A ∈ M, is a measure—meaning it is countably additive on disjoint unions of sets in M. Moreover, this measure is complete: any of a (A ∈ M with μ(A) = 0) is also in M with measure zero. The σ-algebra M is closed under complements (if A ∈ M, then X \ A ∈ M) and countable unions (if {A_n} ⊆ M, then ∪{n=1}^∞ A_n ∈ M), which follows from the splitting property applied iteratively. For disjoint measurable sets {A_n} ∈ M, countable additivity holds: μ(∪{n=1}^∞ A_n) = ∑_{n=1}^∞ μ(A_n), restoring the desired property absent in the full . This restriction is unique in the sense that any measure extending a premeasure on a will coincide with this construction on the generated σ-algebra, as guaranteed by the extension theorem. In practice, this framework underlies the construction of , where the derived from interval lengths restricts to the standard Lebesgue measure on the Borel completed with null sets. The approach, introduced by in 1914, generalizes length to higher dimensions and arbitrary spaces, providing a foundational tool for and probability.