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Riesz–Markov–Kakutani representation theorem

The is a of measure theory and , asserting that for a locally compact X, every positive linear functional T on the space C_c(X) of continuous real-valued functions with compact support on X (equipped with the inductive limit topology) corresponds uniquely to against a \mu on X, meaning T(f) = \int_X f \, d\mu for all f \in C_c(X). This representation ensures the measure \mu is positive, regular (inner and outer regular), and locally finite, capturing the essence of on non-compact spaces in a duality framework. The theorem builds on earlier results, with Frigyes Riesz establishing the case for the unit interval [0,1] in 1909, Andrey Markov extending it to non-compact open subsets of \mathbb{R}^n in 1938, and Shizuo Kakutani generalizing it to compact Hausdorff spaces in 1941, culminating in the full version for arbitrary locally compact Hausdorff spaces. Originally formulated for real scalars, it extends naturally to complex-valued functions via the real and imaginary parts, yielding complex Radon measures. A variant for compact Hausdorff spaces replaces C_c(X) with the full space C(X) of continuous functions (under the sup norm) and yields finite regular Borel measures, highlighting the theorem's adaptability to bounded settings. This result is pivotal for abstract , as it identifies the topological of C_c(X) with the space of measures, providing a constructive link between functionals and geometric objects like measures, which underpins the construction of on locally compact groups and on \mathbb{R}^n. It also facilitates proofs of for measures via the functional representation and supports extensions to vector measures or signed measures by . The theorem's regularity condition ensures measures behave well with respect to compact and open sets, making it indispensable in probability (e.g., representing expectations) and operator algebras (e.g., states on C^*-algebras).

Background Concepts

Locally Compact Hausdorff Spaces

A locally compact X is a in which every point admits a compact neighborhood and that satisfies the , meaning any two distinct points possess disjoint open neighborhoods. This structure combines local , which guarantees the existence of relatively compact open sets around each point, with the Hausdorff condition, ensuring a robust separation of points essential for many topological constructions. Key properties of locally compact Hausdorff spaces include the existence of a one-point compactification, or Alexandrov extension, which embeds X as a dense open subset of a compact by adjoining a single . Additionally, under the assumption of second countability, such spaces are \sigma-compact, admitting a countable cover by compact subsets, which facilitates exhaustive coverings and aids in global analysis. These attributes make locally compact Hausdorff spaces particularly amenable to extensions and approximations in . In the Riesz–Markov–Kakutani representation theorem, the locally compact Hausdorff setting is crucial as it ensures that continuous functions on X exhibit controlled suitable for , with compact subsets naturally supporting finite measures in the representation. examples include spaces \mathbb{R}^n, where balls serve as compact neighborhoods, and finite-dimensional smooth manifolds, which inherit this structure from their charts. In contrast, non-Hausdorff spaces, such as those with the indiscrete (where only the and the full space are open), fail the and thus cannot qualify as locally compact Hausdorff.

Function Spaces and Linear Functionals

In the context of a locally compact Hausdorff X, the space C_c(X) consists of all continuous real-valued functions f: X \to \mathbb{R} that have compact support, meaning the set \{x \in X : f(x) \neq 0\} is compact. This space is equipped with the inductive limit topology, obtained as the direct limit of the sup-norm topologies on the subspaces of functions supported in each compact subset of X. On \sigma-compact spaces, C_c(X) is dense in L^p(X) for $1 \leq p < \infty with respect to the L^p norm. The space C_0(X) is defined as the closure of C_c(X) in the supremum norm, comprising all continuous functions f: X \to \mathbb{R} that vanish at infinity, i.e., for every \epsilon > 0, the set \{x \in X : |f(x)| \geq \epsilon\} is compact. Equivalently, \lim_{|x| \to \infty} f(x) = 0 uniformly outside compact subsets. Under the supremum norm, C_0(X) forms a . A positive linear functional on C_c(X) is a \psi: C_c(X) \to \mathbb{R} such that \psi(f) \geq 0 whenever f \geq 0 ; such functionals are continuous with respect to the inductive limit . Continuous linear functionals on C_0(X) are bounded linear maps \psi: C_0(X) \to \mathbb{R}, with \|\psi\| = \sup_{\|f\|_\infty \leq 1} |\psi(f)|. These function spaces serve as models for functions in , facilitating the interplay between topological properties of X and analytic structures like and duality.

Core Theorems

Positive Functionals on C_c(X)

The Riesz–Markov representation theorem asserts that for a locally compact X, every positive linear functional \psi: C_c(X) \to \mathbb{R} is represented by against a unique positive \mu on X satisfying \psi(f) = \int_X f \, d\mu for all f \in C_c(X). The representing measure \mu possesses several key properties that ensure its suitability for this representation. It is finite on compact subsets, meaning \mu(K) < \infty for every compact set K \subset X. Additionally, \mu is outer regular, so for every E \subset X, \mu(E) = \inf \{ \mu(U) : U \supset E, \, U \text{ open} \}, and inner regular, so for every U \subset X, \mu(U) = \sup \{ \mu(K) : K \subset U, \, K \text{ compact} \}. The (X, \mathcal{B}(X), \mu), where \mathcal{B}(X) is the Borel \sigma-algebra, is complete: if E \in \mathcal{B}(X), A \subset E, and \mu(E) = 0, then A \in \mathcal{B}(X) and \mu(A) = 0. These regularity conditions distinguish \mu as a when X is \sigma-compact, in which case \mu is locally finite and inner regular on all Borel sets. Uniqueness of \mu follows from its determination by the values of \psi on compact sets and the imposed regularity properties; distinct regular measures agreeing on C_c(X) do not exist under these conditions. Without regularity, however, the representing need not be unique, as seen in pathological spaces like certain uncountable ordinals where multiple measures can induce the same functional. The integral in the representation formula \psi(f) = \int_X f(x) \, d\mu(x) is defined via , constructed by approximating f \in C_c(X) with simple functions through limits, leveraging the to ensure consistency with \psi. This approach aligns the functional's action directly with the measure's properties on compact supports.

Dual of C_0(X)

The continuous of C_0(X), where X is a locally compact and C_0(X) denotes the Banach space of complex-valued continuous functions on X vanishing at equipped with the supremum , consists of all continuous linear functionals \psi: C_0(X) \to \mathbb{C}. By the Kakutani representation theorem, every such \psi admits a unique representation of the form \psi(f) = \int_X f(x) \, d\mu(x) for all f \in C_0(X), where \mu is a complex regular on X. The measure \mu is regular in the sense that it is both outer regular and inner regular: for every Borel set E \subseteq X, \mu(E) = \inf\{\mu(U) : U \supseteq E, \, U \text{ open}\}, and for every open set U \subseteq X, \mu(U) = \sup\{\mu(K) : K \subseteq U, \, K \text{ compact}\}. Moreover, the total variation measure |\mu| satisfies |\mu|(X) = \|\psi\|, where \|\psi\| is the operator norm of \psi, establishing an isometric isomorphism between C_0(X)^* and the space of complex regular Borel measures on X endowed with the total variation norm. In the special case where \psi is real-valued and positive (i.e., \psi(f) \geq 0 whenever f \geq 0), the representing measure \mu is a . For a general complex \psi, the representation arises by decomposing \psi = \psi_1 + i \psi_2 into real and imaginary parts, each corresponding to a real regular via the positive case, and defining \mu = \mu_1 + i \mu_2. The uniqueness of \mu holds under the regularity condition, distinguishing it from non-regular representations; this follows from the density of C_c(X) in C_0(X) and the continuity of \psi. The positive case for real functionals on C_0(X) reduces directly to the representation on the dense C_c(X).

Proof Outlines

Measure Construction for C_c(X)

The construction of the representing measure \mu begins with the positive linear functional \psi: C_c(X) \to \mathbb{R} on the space of continuous real-valued functions with compact support on a locally compact X, as per the Riesz–Markov–Kakutani representation theorem for C_c(X). To define \mu initially on compact subsets, for each compact set K \subset X, set \mu(K) = \sup \{ \psi(f) \mid f \in C_c(X),\ 0 \leq f \leq \chi_K \}, where \chi_K is the of K. This supremum is well-defined and finite because \psi is positive and locally bounded: for any compact K, the restriction of \psi to functions supported in K is bounded above by \psi(g) for some g \in C_c(X) with $0 \leq g \leq 1 on K and support in a neighborhood of K. Moreover, this definition ensures additivity on disjoint compact sets; if K_1 and K_2 are disjoint compacts, functions approximating \chi_{K_1} and \chi_{K_2} can be combined without overlap using the and positivity of \psi, yielding \mu(K_1 \cup K_2) = \mu(K_1) + \mu(K_2). Next, extend \mu to open sets and then to the Borel \sigma-algebra. For an open set U \subset X, define \mu(U) = \sup \{ \mu(K) \mid K \subset U,\ K compact \}, which establishes inner regularity. To reach all Borel sets, first construct an outer measure \mu^* on the power set of X by \mu^*(A) = \inf \{ \sum \mu(U_i) \mid A \subset \bigcup U_i,\ U_i open \} for arbitrary A \subset X. The collection of \mu^*-measurable sets then forms a \sigma-algebra containing the Borel sets, and restricting \mu^* to this algebra yields a measure \mu on the Borel \sigma-algebra via the Carathéodory extension theorem, as the premeasure on open sets (or the semiring of differences of opens) is \sigma-additive due to the local compactness ensuring controlled approximations. Alternatively, the Daniell integral approach can be used by viewing \psi as defining an integral over simple functions generated by partitions of unity, extending to the Borel algebra while preserving the representing property. This extension maintains outer regularity by construction and achieves inner regularity on Borel sets without initially assuming \sigma-compactness, relying instead on the exhaustion of open sets by compacta in locally compact spaces. To verify the representation \psi(f) = \int_X f \, d\mu for all f \in C_c(X), approximate f by simple functions. Since f has compact support, partition the range of f into intervals and use continuous functions f_n such that $0 \leq f_n \uparrow f pointwise, with \mathrm{supp}(f_n) compact. By the for the measure \mu, \int f_n \, d\mu \to \int f \, d\mu, and of \psi gives \psi(f_n) \to \psi(f); moreover, each \psi(f_n) equals \int f_n \, d\mu by additivity over the partition sets, whose measures are defined consistently. For characteristic functions of compacts, the approximation directly ties back to the supremum definition. Positivity of \mu follows immediately, as \psi(f) \geq 0 for f \geq 0 implies \mu(E) \geq 0 for Borel E, and local finiteness holds since \mu(K) < \infty for compact K.

Extension to C_0(X)

The extension of the representation theorem from the space of compactly supported continuous functions C_c(X) to the space of continuous functions vanishing at infinity C_0(X) on a locally compact X relies on the density of C_c(X) in C_0(X) with respect to the supremum norm. Specifically, C_c(X) is dense in C_0(X), meaning that for any f \in C_0(X), there exists a \{f_n\} \subset C_c(X) such that \|f_n - f\|_\infty \to 0 as n \to \infty. Given a continuous linear functional \psi: C_0(X) \to \mathbb{C}, its restriction \psi|_{C_c(X)} is a continuous linear functional on C_c(X), which, by the prior construction for C_c(X), is represented by against a unique regular \mu on X such that \psi(g) = \int_X g \, d\mu for all g \in C_c(X). For a general continuous \psi, first consider the case of positive real-valued functionals, where \psi(f) \geq 0 for f \geq 0. Here, the measure \mu obtained from the restriction is positive, and the representation extends to all of C_0(X) by continuity: for f \in C_0(X), define \int_X f \, d\mu = \lim_{n \to \infty} \int_X f_n \, d\mu = \lim_{n \to \infty} \psi(f_n) = \psi(f), where \{f_n\} approximates f uniformly from C_c(X). This limit exists and is independent of the approximating sequence due to the uniform convergence and the bounded variation of \mu. For non-positive real functionals, decompose \psi = \psi^+ - \psi^-, where \psi^+ and \psi^- are positive functionals defined via the Hahn-Jordan decomposition, yielding \psi(f) = \int_X f \, d\mu^+ - \int_X f \, d\mu^- with positive regular measures \mu^+ and \mu^-; the total variation measure |\mu| = \mu^+ + \mu^- ensures regularity by inheriting inner and outer regularity from each component. In the complex case, \psi = \operatorname{Re} \psi + i \operatorname{Im} \psi, where the real and imaginary parts are real linear functionals represented as differences of positive measures, leading to a regular \mu via \mu = \mu_{\operatorname{Re}} + i \mu_{\operatorname{Im}}, with the defined componentwise. Alternatively, the total variation approach constructs |\psi|, the positive functional given by |\psi|(f) = \sup \{ |\sum z_k \psi(f_k)| : \sum |z_k| = 1, f_k \geq 0, \sum f_k \leq f \}, which yields a positive measure |\mu| such that \psi(f) = \int_X f \, d\mu and the representation holds by density arguments. The Hahn-Banach theorem facilitates this decomposition by extending positive parts while preserving norms, ensuring the extension remains continuous. Continuity of the extended representation is verified by the norm equality \|\psi\| = |\mu|(X), achieved through approximations: for f \in C_0(X) with \|f\|_\infty \leq 1, |\psi(f)| \leq \sup_n |\psi(f_n)| \leq |\mu|(X), and conversely, using partitions of or Urysohn functions to approximate the characteristic of sets where |\mu| concentrates, yielding the reverse inequality via uniform limits. Regularity of \mu follows from the density: for open U, \mu(U) = \sup \{ \mu(K) : K \subset U, K \text{ compact} \} holds by approximating indicator functions of U uniformly with compactly supported functions, and similarly for closed sets using complements. A key tool in these approximations on compact subsets is the Stone-Weierstrass theorem, which guarantees dense algebraic subalgebras (e.g., polynomials on \mathbb{R}^n) in C(K) for compact K, extended globally by localizing via partitions of to handle functions in C_0(X).

Historical Context

Riesz's Foundational Work

Frigyes Riesz's contributions to in the early 1900s were instrumental in advancing the study of integral equations and laying the foundations for theory, amid a broader effort to generalize classical analysis to infinite-dimensional settings. Motivated by David Hilbert's work on integral equations, Riesz sought to characterize linear operations on function spaces, building on emerging ideas in measure and integration theory following Henri Lebesgue's 1902 dissertation. In this context, Riesz's 1909 paper, published in French as "Sur les opérations fonctionnelles linéaires" in the Comptes rendus hebdomadaires des séances de l'Académie des sciences, introduced a key representation result for a specific class of functionals. The core result of Riesz's work states that every positive linear functional on the space of continuous real-valued functions C([0,1]) on the compact interval [0,1] can be represented as an against a positive measure derived from a non-decreasing of . This representation takes the form of a Stieltjes , where the functional Λ applied to a f yields ∫_{[0,1]} f(x) dα(x), with α a of that ensures the measure is positive and finitely additive. Riesz's proof relied on constructing such an α through approximations using linear functions and analyzing difference quotients to establish the existence of upper right Dini derivatives, thereby linking the functional directly to a integrator without invoking general measure theory. This innovation represented a significant shift from Riemann integration toward a more robust framework akin to , as it allowed functionals to be expressed via measures supported on functions of , unifying pointwise with forms. By the within the classical setting of the unit interval, Riesz bridged concrete with abstract , highlighting how ensures the 's well-definedness and positivity preservation. Despite its elegance, Riesz's theorem was confined to the compact domain [0,1], addressed only real-valued continuous functions, and operated without the machinery of or in abstract spaces. The 1909 paper served primarily as an announcement, with a fuller proof appearing in Riesz's subsequent work on equations. Riesz's achievement provided the initial blueprint for abstract representations, influencing the evolution of duality in function spaces and paving the way for broader applications in .

Markov and Kakutani Contributions

In 1938, extended the scope of Riesz's representation theorem by addressing positive linear functionals on the space of continuous real-valued functions with compact support, C_c(X), defined on non-compact open subsets of \mathbb{R}^n. Markov demonstrated that every such positive linear functional corresponds uniquely to integration with respect to a positive on X, where regularity ensures that the measure of open sets equals the infimum of measures of open covers and the measure of compact sets equals the supremum of measures of compact subsets. This work, published as "On mean values and exterior densities" in the journal Recueil Mathématique de la Société Mathématique de Moscou (new series), introduced techniques for handling non-compact spaces through outer and interior densities, thereby generalizing Riesz's compact-case result to broader topological settings in . Markov's contribution emphasized the role of positivity in ensuring the existence of representing measures without requiring a priori assumptions in the , though such functionals are continuous in the inductive limit on C_c(X). His approach facilitated the of measures that are finite on compact sets and countably additive on Borel sets, laying groundwork for applications in and over unbounded domains. In 1941, further generalized the theorem to characterize the of the C(X) of continuous functions on a compact X (under the sup norm). Kakutani proved that every continuous linear functional on C(X) is uniquely represented by against a regular on X. This result appeared in his paper "Concrete representation of abstract (M)-spaces (A characterization of the space of continuous functions)" in the Annals of Mathematics, where he linked the structure to abstract (M)-spaces, providing an lattice to spaces of continuous functions. Originally for real scalars, the result extends to complex-valued functions via real and imaginary parts, yielding complex measures. Kakutani's advances incorporated the general compact Hausdorff setting and focused on the to ensure boundedness, thus bridging the theorem to general duality and . His work completed the transition from specific real positive functionals to full continuous duals in compact settings. The Riesz–Markov–Kakutani reflects these incremental developments: Riesz's initial compact real case on [0,1], Markov's positive extension to compact-support functions on open subsets of \mathbb{R}^n, and Kakutani's comprehensive version for continuous functionals on compact Hausdorff spaces. The full version for arbitrary locally compact Hausdorff spaces was synthesized in subsequent works, such as those by Bourbaki in the . Post-World War mathematical literature, including foundational texts in , solidified the theorem's unified form and widespread adoption.