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Measurable function

In measure theory, a measurable function is a mapping between two measurable spaces that preserves measurability in the sense that the preimage of every measurable set in the codomain is a measurable set in the domain.^[1] This concept generalizes the notion of continuity from topology, where continuous functions pull back open sets to open sets, to the broader framework of measure spaces where measurable functions pull back measurable sets to measurable sets.^[1] Measurable functions form the foundation for integration theory, enabling the definition and study of integrals over spaces equipped with measures, such as the Lebesgue integral on the real line.^[2] For real-valued functions f: X \to \mathbb{R} defined on a measurable space (X, \mathcal{A}), measurability is typically with respect to the Borel \sigma-algebra on \mathbb{R}, and it is equivalent to the condition that for every real number a, the set \{x \in X : f(x) > a\} is in \mathcal{A}.^[3] A key subclass consists of simple functions, which are finite linear combinations of indicator functions of measurable sets and serve as building blocks for approximating more general measurable functions.^[1] Measurable functions exhibit several important algebraic and limit properties that facilitate their use in analysis. The sum and product of two measurable functions are measurable, as are scalar multiples, though the class is not closed under composition unless the outer function is continuous.^[3] Pointwise limits of sequences of measurable functions are also measurable, which underpins theorems like the monotone convergence theorem for integration.^[1] In complete measure spaces, functions that agree almost everywhere with measurable functions are themselves measurable, allowing for equivalence classes modulo sets of measure zero.^[1] These properties ensure that measurable functions are robust under the operations central to probability and analysis, extending beyond continuous functions to handle phenomena like discontinuities on sets of measure zero.^[4]

Foundations of Measure Theory

Measure Spaces

A measure space is a basic structure in measure theory, consisting of a triple (X, \Sigma, \mu), where X is an arbitrary set, \Sigma is a \sigma-algebra on X (the collection of measurable sets), and \mu is a measure defined on \Sigma.^[5]^[6] A measure \mu is a non-negative extended real-valued set function on \Sigma satisfying \mu(\emptyset) = 0 and countable additivity: for any countable collection of pairwise disjoint sets \{A_n\}_{n=1}^\infty \subset \Sigma, \mu\left(\bigcup_{n=1}^\infty A_n\right) = \sum_{n=1}^\infty \mu(A_n).^[7]^[5] This ensures that \mu assigns a consistent notion of "size" or "volume" to the measurable subsets of X, generalizing concepts like length, area, and probability.^[6] Prominent examples include the Lebesgue measure on \mathbb{R}^n, which assigns to each measurable set its "volume" in the sense of integration, such as the length of intervals in \mathbb{R}.^[8] The counting measure on a countable set X defines \mu(A) as the cardinality of A if A is finite and \infty otherwise, effectively counting elements.^[6] Another example is the Dirac measure \delta_x at a point x \in X, where \delta_x(A) = 1 if x \in A and $0 otherwise, concentrating all mass at x.^[7] These structures provide the foundational framework for defining integrals and analyzing functions by quantifying the sizes of sets in a rigorous, additive manner.^[5]

Measurable Sets and Sigma-Algebras

In measure theory, a σ-algebra on a set X is a collection \mathcal{F} of subsets of X that includes the empty set \emptyset and X itself, and is closed under complementation and countable unions. Specifically, if A \in \mathcal{F}, then its complement X \setminus A \in \mathcal{F}, and if A_1, A_2, \dots \in \mathcal{F}, then \bigcup_{n=1}^\infty A_n \in \mathcal{F}.^[9] These axioms ensure that \mathcal{F} forms a Boolean algebra extended to countable operations, providing the foundational structure for defining measurable sets.^[10] Key properties of σ-algebras include closure under countable intersections, which follows from De Morgan's laws applied to complements of unions, and closure under finite unions and intersections as special cases of the countable versions. Additionally, σ-algebras are monotone in the sense that if A \subseteq B with A, B \in \mathcal{F}, then B \setminus A \in \mathcal{F} via differences, though not all subsets between elements need be included. The closure under countable unions ensures that arbitrary countable collections of sets in \mathcal{F} yield unions still in \mathcal{F}, without requiring disjointness. Measures are defined as countably additive set functions on σ-algebras and satisfy subadditivity for general countable unions.^[9]^[10] The σ-algebra generated by a family of sets \mathcal{B} \subseteq \mathcal{P}(X) (the power set of X) is the smallest σ-algebra containing \mathcal{B}, obtained as the intersection of all σ-algebras that include \mathcal{B}. This generated σ-algebra, denoted \sigma(\mathcal{B}), captures the minimal extension needed for measurability starting from \mathcal{B}. For example, the power set \mathcal{P}(X) itself is a σ-algebra, as it includes all subsets and is closed under all required operations, serving as the largest possible σ-algebra on X.^[9]^[10] In topological spaces, the Borel σ-algebra \mathcal{B}(X) is generated by the open sets of the topology, making it the smallest σ-algebra containing all open subsets; this construction is fundamental for Borel measurability in \mathbb{R}^n or more general spaces. The Lebesgue σ-algebra on \mathbb{R} extends the Borel σ-algebra by completion with respect to Lebesgue measure, incorporating all subsets of Borel null sets to form a larger collection while preserving measure properties.^[11]^[12]

Definition and Variations

Formal Definition

In measure theory, a measurable function is defined between two measurable spaces. Let (X, \Sigma_X) and (Y, \Sigma_Y) be measurable spaces, where \Sigma_X and \Sigma_Y are \sigma-algebras on sets X and Y, respectively. A function f: X \to Y is measurable if the preimage f^{-1}(E) \in \Sigma_X for every set E \in \Sigma_Y.^[1] This condition ensures that the function preserves the structure of measurability under inverse images. A key property underlying this definition is the behavior of preimages under set operations. For a countable collection of sets \{E_i\}_{i \in I} \subset \Sigma_Y, the preimage satisfies f^{-1}\left( \bigcup_{i \in I} E_i \right) = \bigcup_{i \in I} f^{-1}(E_i), which aligns with the closure properties of \sigma-algebras.^[1] For real-valued functions, the definition specializes to the Borel \sigma-algebra on \mathbb{R}. Consider a measurable space (X, \Sigma) and a function f: X \to \mathbb{R}. The function f is measurable if f^{-1}(B) \in \Sigma for every Borel set B \in \mathcal{B}(\mathbb{R}), where \mathcal{B}(\mathbb{R}) is generated by the open intervals of \mathbb{R}.^[1] An equivalent condition is that f^{-1}((a, \infty)) \in \Sigma for all a \in \mathbb{R}, since the half-lines generate the Borel \sigma-algebra.^[1] This framework extends to the extended real line \overline{\mathbb{R}} = \mathbb{R} \cup \{\pm \infty\}. A function f: X \to \overline{\mathbb{R}} is measurable if f^{-1}(B) \in \Sigma for every Borel set B \in \mathcal{B}(\overline{\mathbb{R}}), or equivalently, if \{x \in X : f(x) < b\} \in \Sigma for all b \in \overline{\mathbb{R}}.^[1] For complex-valued functions, f: X \to \mathbb{C}, measurability holds if and only if both the real part \operatorname{Re} f and the imaginary part \operatorname{Im} f are measurable as real-valued functions.^[13]

Terminology and Contextual Usage

In measure theory, the term "measurable function" typically refers to a function f: X \to \mathbb{R} defined on a measurable space (X, \mathcal{A}) such that the preimage f^{-1}(B) belongs to \mathcal{A} for every Borel set B \subseteq \mathbb{R}.^[1] A key distinction arises between Borel measurability and Lebesgue measurability on \mathbb{R}^n. A function is Borel measurable if the preimage of every Borel set is a Borel set, whereas it is Lebesgue measurable if the preimage of every Borel set is Lebesgue measurable; the latter class is larger because the Lebesgue \sigma-algebra includes all Borel sets plus certain null sets and their complements.^[1] In probability theory, measurable functions play a central role as random variables, defined on a probability space (\Omega, \mathcal{F}, P) and taking values in \mathbb{R} (or more generally, a measurable space), such that the preimage of every Borel set in \mathbb{R} lies in \mathcal{F}.^[14] This usage allows probabilities to be assigned to events of the form \{ \omega \in \Omega : X(\omega) \in B \}, where X is the random variable, facilitating the analysis of stochastic processes and expectations.^[14] Historically, the concept of measurability was formalized by Constantin Carathéodory in 1914 through his extension theorem, which constructs measures from outer measures and defines measurable sets via a splitting condition, thereby linking measurability directly to the foundations of integration theory.^[15] In more advanced contexts, such as Bochner integration on Banach spaces, variations like "weakly measurable" and "strongly measurable" functions emerge. A function f with values in a Banach space X is weakly measurable if its composition with every continuous linear functional from the dual space X' is scalar measurable, while it is strongly measurable if it is the almost everywhere pointwise limit of simple functions with values in X.^[16] Additionally, two functions are often considered equivalent if they agree almost everywhere with respect to the underlying measure, preserving measurability properties in L^p spaces.^[1]

Properties of Measurable Functions

Algebraic and Arithmetic Properties

Measurable functions exhibit closure under various algebraic and arithmetic operations, forming a vector space over the real or complex numbers when restricted to finite-valued functions. Specifically, if f and g are measurable functions from a measurable space (X, \mathcal{M}) to \mathbb{R}, then their pointwise sum f + g is also measurable. This follows from expressing the preimage (f + g)^{-1}((-\infty, b)) = \bigcup_{\substack{q, r \in \mathbb{Q} \\ q + r < b}} f^{-1}((-\infty, q)) \cap g^{-1}((-\infty, r)), which is a countable union of intersections of measurable sets, hence measurable.^[1] Similarly, scalar multiplication preserves measurability: for any constant k \in \mathbb{R} and measurable f: X \to \mathbb{R}, the function k f is measurable. If k > 0, then (k f)^{-1}((-\infty, b)) = f^{-1}((-\infty, b/k)), which is measurable; the case k < 0 follows analogously by reflecting the preimage, and k = 0 yields the constant zero function, which is measurable.^[1] More generally, the set of finite-valued measurable functions on (X, \mathcal{M}) is closed under pointwise addition and scalar multiplication, making it a vector space.^[17] Composition of measurable functions also preserves measurability. If f: (X, \mathcal{M}_X) \to (Y, \mathcal{M}_Y) and g: (Y, \mathcal{M}_Y) \to (Z, \mathcal{M}_Z) are measurable, then g \circ f: X \to Z is measurable, since for any E \in \mathcal{M}_Z, (g \circ f)^{-1}(E) = f^{-1}(g^{-1}(E)), and both g^{-1}(E) \in \mathcal{M}_Y and its preimage under f are in \mathcal{M}_X.^[1] This property extends to cases where the codomain is equipped with the Borel \sigma-algebra, such as when composing with Borel measurable functions on \mathbb{R}.^[18] The class of measurable functions is stable under pointwise limits. If \{f_n\} is a sequence of measurable functions from (X, \mathcal{M}) to \overline{\mathbb{R}} (the extended reals) converging pointwise to f: X \to \overline{\mathbb{R}}, then f is measurable. This holds because f can be expressed using limsup and liminf operations, each of which is measurable as a pointwise supremum or infimum of measurable functions: for instance, \limsup_{n \to \infty} f_n = \inf_{n \geq 1} \sup_{k \geq n} f_k, and such countable suprema and infima preserve measurability.^[1] Consequently, uniform limits, monotone limits, and other sequential limits of measurable functions remain measurable, facilitating approximations in integration theory.^[19]

Continuity and Topological Relations

In measure theory, the relationship between continuity and measurability highlights how topological regularity aligns with σ-algebra structures. A continuous function f: X \to Y between topological spaces equipped with their Borel σ-algebras is Borel measurable, as the preimage f^{-1}(U) of any open set U \subseteq Y is open in X, hence Borel.^[20] This follows directly from the definition of the Borel σ-algebra generated by open sets and the continuity condition that preserves openness under inverse images.^[20] Lusin's theorem further bridges measurability and continuity by asserting that every finite-valued Borel measurable function on a compact metric space is continuous on a subset whose complement has arbitrarily small measure.^[21] Specifically, for any \epsilon > 0, there exists a compact set K with measure at least \mu(X) - \epsilon such that f restricted to K is continuous.^[21] This result underscores that Borel measurability implies near-continuity in the measure-theoretic sense on compact domains. Extending to Lebesgue measurable functions on \mathbb{R}, the Denjoy-Young-Saks theorem implies that such functions are approximately continuous almost everywhere.^[22] Approximate continuity at a point x means that for every \epsilon > 0, the density of f^{-1}(B(f(x), \epsilon)) at x is 1, where density is taken with respect to Lebesgue measure; this holds except on a set of measure zero, reflecting the theorem's classification of Dini derivatives almost everywhere.^[22] Regarding Baire category, measurable functions avoid certain pathological discontinuities prevalent in the first category. In particular, every Lebesgue measurable function coincides almost everywhere with a function of Baire class 2, ensuring that its discontinuities do not form a comeager set and thus sidestep the dense-in-itself irregularities that plague non-measurable examples.^[23] This topological regularity complements measure-theoretic properties, as the Baire category theorem guarantees that residual sets of discontinuities are meager, aligning measurable functions with "generic" continuous-like behavior in complete metric spaces.^[24]

Classes of Measurable Functions

Simple and Step Functions

In measure theory, a simple function on a measurable space (X, \mathcal{A}) is a measurable function \phi: X \to \mathbb{R} that can be expressed as a finite linear combination of indicator functions of measurable sets, specifically \phi = \sum_{i=1}^n c_i \chi_{E_i}, where each c_i \in \mathbb{R}, the E_i \in \mathcal{A} are measurable sets (often taken to be disjoint in a canonical representation), and \chi_{E_i} is the characteristic function of E_i.^[1] This form ensures that \phi takes only finitely many values, and every simple function admits such a representation with finitely many nonzero c_i.^[25] Simple functions form an algebra under pointwise addition and multiplication, preserving measurability and the finite-range property.^[25] In the specific context of functions on \mathbb{R} (or intervals thereof) equipped with the Lebesgue \sigma-algebra, a step function is a simple function that is constant on a finite collection of subintervals, meaning the sets E_i are finite unions of intervals.^[26] Thus, every step function is simple, as intervals are Lebesgue measurable, but the converse does not hold in general, since simple functions can involve arbitrary measurable sets rather than just intervals.^[26] Step functions arise naturally in Riemann integration as piecewise constant approximations but gain broader utility in Lebesgue theory for their role in foundational constructions.^[27] Simple functions play a central role in defining the Lebesgue integral on \mathbb{R}^d. For a nonnegative simple function \phi = \sum_{i=1}^n c_i \chi_{E_i} with c_i \geq 0 and Lebesgue measurable E_i, the Lebesgue integral is defined as \int_{\mathbb{R}^d} \phi \, dm = \sum_{i=1}^n c_i m(E_i), where m denotes Lebesgue measure; this extends linearly to general simple functions and provides the basis for integrating arbitrary nonnegative measurable functions via monotone approximation by simple functions.^[27] Under Lebesgue measure on \mathbb{R}^d, the simple functions are dense in the L^p spaces for $1 \leq p < \infty, meaning that for any f \in L^p(\mathbb{R}^d) and \epsilon > 0, there exists a simple function \phi such that \|f - \phi\|_p < \epsilon.^[28] This density follows from the pointwise approximation of measurable functions by increasing sequences of simple functions and the dominated convergence theorem applied to the p-th powers.^[28]

Borel and Lebesgue Measurable Functions

In real analysis, a function f: \mathbb{R}^n \to \mathbb{R} is Borel measurable if the preimage f^{-1}(B) is a Borel set for every Borel set B \subseteq \mathbb{R}, where the Borel \sigma-algebra on \mathbb{R}^n is generated by the open sets.^[1] This class includes all continuous functions, as the preimage under a continuous map of an open set (and hence any Borel set) is open and thus Borel.^[1] Polynomials, being continuous, are therefore Borel measurable.^[1] A function f: \mathbb{R}^n \to \mathbb{R} is Lebesgue measurable if f^{-1}(B) belongs to the Lebesgue \sigma-algebra for every Borel set B \subseteq \mathbb{R}.^[1] The Lebesgue \sigma-algebra is the completion of the Borel \sigma-algebra with respect to Lebesgue measure, consisting of all sets of the form A \cup N or A \setminus N, where A is Borel and N is contained in a Borel set of measure zero.^[12] Every Borel measurable function is Lebesgue measurable, since Borel sets are Lebesgue measurable, but the converse does not hold.^[1] However, every Lebesgue measurable function equals a Borel measurable function almost everywhere with respect to Lebesgue measure.^[1] Lusin's theorem characterizes Lebesgue measurable functions through their near-continuity. Specifically, for a Lebesgue measurable set E \subseteq \mathbb{R}^d and a function f: E \to \mathbb{C}, f is measurable if and only if for every \varepsilon > 0, there exists a compact set K \subseteq E with m(E \setminus K) < \varepsilon such that f restricted to K is continuous.^[29] Lusin spaces generalize this property to broader topological settings. A topological measure space (X, \mu) is a Lusin space if \mu(X) < \infty and every real-valued \mu-measurable function on X equals a continuous function \mu-almost everywhere.^[30] In such spaces, the topology ensures that measurability implies near-continuity for all finite Borel measures.^[31]

Non-Measurable Functions

Existence and Axiom of Choice

The existence of non-measurable functions is closely tied to the existence of non-measurable sets, as the characteristic function of a non-measurable set is itself non-measurable with respect to the Lebesgue σ-algebra.^[32] In 1905, Giuseppe Vitali provided the first construction of a non-Lebesgue measurable subset of the real line, relying implicitly on the axiom of choice to select representatives from equivalence classes under rational translations.^[32] This marked the historical recognition that the axiom of choice enables the formation of sets outside the Lebesgue measurable class. The axiom of choice (AC), which asserts that for any collection of nonempty sets there exists a choice function selecting one element from each, implies the existence of non-Lebesgue measurable subsets of \mathbb{R}^n for any n \geq 1.^[32] Specifically, AC allows the construction of a set that intersects every interval in a way that defies additive measure properties, leading to non-measurable indicator functions.[](https://e.math.cornell.edu/people/belk/measure theory/NonMeasurableSets.pdf) A striking illustration of this implication is the Banach-Tarski paradox, proved in 1924, which uses AC to decompose the unit ball in \mathbb{R}^3 into finitely many pieces that can be reassembled via rigid motions into two copies of the original ball. These pieces are non-measurable sets, as any measurable decomposition preserving Lebesgue measure would violate volume additivity, thus yielding non-measurable functions when considering their indicators or transformations. The necessity of AC for non-measurability is underscored by results in set theory without choice. In 1970, Robert M. Solovay constructed a model of Zermelo-Fraenkel set theory (ZF) plus the axiom of dependent choices (DC) in which every set of real numbers is Lebesgue measurable, demonstrating that the existence of non-measurable sets (and hence non-measurable functions) is not provable in ZF alone and requires AC. This model, built using an inaccessible cardinal in the base theory, preserves DC for countable choice while ensuring all subsets of \mathbb{R} belong to the Lebesgue σ-algebra, highlighting AC's role in permitting pathological non-measurable phenomena.

Constructions and Examples

One prominent construction of a non-measurable set, and thus a non-measurable function, is the Vitali set, introduced by Giuseppe Vitali in 1905. Consider the real numbers \mathbb{R} modulo the rationals \mathbb{Q}, forming equivalence classes where x \sim y if x - y \in \mathbb{Q}. Using the axiom of choice, select one representative from each equivalence class intersected with the interval [0,1) to form the set V \subset [0,1). This set V is dense in [0,1) and intersects every subinterval of [0,1) with positive length.^[33] To see that V is non-Lebesgue measurable, consider the countable collection of disjoint sets V_n = (V + r_n) \cap [0,1), where \{r_n\} enumerates \mathbb{Q} \cap (-1,1). These sets are measurable with the same measure as V (by translation invariance), and their union is [0,1). If \mu(V) > 0, then \mu([0,1)) = \sum \mu(V_n) = \infty > 1, a contradiction. If \mu(V) = 0, then \mu([0,1)) = 0 < 1, another contradiction. Thus, V is non-measurable, and its indicator function $1_V: \mathbb{R} \to \{0,1\}, defined by $1_V(x) = 1 if x \in V and 0 otherwise, is non-measurable since the preimage of \{1\} is V, which fails the measurability condition for Borel sets.^[33] Another construction yields non-measurable functions via a Hamel basis for \mathbb{R} as a vector space over \mathbb{Q}. The axiom of choice guarantees the existence of such a basis B, an uncountable set where every real number has a unique finite linear combination representation with rational coefficients from elements of B. Define a linear functional f: \mathbb{R} \to \mathbb{R} by assigning arbitrary values to basis elements (e.g., f(b) = 1 for all b \in B) and extending linearly. This f is \mathbb{Q}-linear but discontinuous everywhere, hence non-measurable, as continuous linear functionals on \mathbb{R} are merely multiplication by a constant.^[34] The Sierpiński–Mazurkiewicz paradox provides further examples of non-measurable sets through paradoxical decompositions in the plane. In 1914, Sierpiński and Mazurkiewicz showed that there exists a set E \subset \mathbb{R}^2 that can be partitioned into two disjoint subsets E_1 and E_2, each congruent to E via isometries, implying E is non-measurable under Lebesgue measure since measurable sets cannot satisfy such equidecomposability without measure preservation. The indicator functions $1_{E_1} and $1_{E_2} are likewise non-measurable. This construction relies on the axiom of choice to select representatives in a decomposition involving free groups acting on the plane.^[35]

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