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Lebesgue integral

The Lebesgue integral is a generalization of the Riemann integral, introduced by French mathematician Henri Lebesgue in his 1902 doctoral dissertation Intégrale, longueur, aire, that defines the integral of a function with respect to Lebesgue measure on the real line or more general measure spaces.^[1] Unlike the Riemann integral, which approximates areas by partitioning the domain into intervals and summing products of widths and function values, the Lebesgue integral partitions the range of the function into levels and integrates by summing the measures of the preimages multiplied by those levels, enabling integration over a much broader class of functions, including those discontinuous on sets of positive measure.^[2]^[3] Lebesgue's work addressed limitations of the Riemann integral exposed in the late 19th century, particularly in the study of Fourier series and the pointwise convergence of sequences of functions, where functions like the Dirichlet function—equal to 1 on rational numbers and 0 on irrationals—are not Riemann integrable over [0,1] but have Lebesgue integral 0 since the rationals have measure zero.^[4]^[5] This innovation, rooted in earlier ideas of measure and length from mathematicians like Georg Cantor and Émile Borel, established the foundations of modern measure theory and transformed real analysis by providing tools to handle "pathological" functions that arise in advanced mathematics.^[2]^[6] Among its notable properties, the Lebesgue integral satisfies linearity (∫(af + bg) = a∫f + b∫g for constants a, b and integrable f, g), monotonicity (if 0 ≤ f ≤ g then ∫f ≤ ∫g), and countable additivity over disjoint sets, while theorems such as the monotone convergence theorem and dominated convergence theorem allow limits to pass inside the integral under mild conditions, facilitating proofs in probability theory, functional analysis, and partial differential equations.^[7]^[8] It also coincides with the Riemann integral whenever the latter exists, ensuring compatibility with classical results, and forms the basis for L^p spaces, which are essential in studying norms and convergence in infinite-dimensional settings.^[3]^[9]

Overview

Historical context and motivation

Henri Léon Lebesgue was born on June 28, 1875, in Beauvais, France, to a modest family; his father was a typesetter in a printing shop. He excelled in his early education, entering the École Normale Supérieure in Paris in 1894 after preparatory studies at local lycées. Graduating in 1897, Lebesgue briefly taught at lycées in Rennes and Besançon before taking a position at the Lycée Central in Nancy from 1899 to 1902, where he developed the core ideas for his groundbreaking work. In 1902, at age 27, he submitted his doctoral thesis, Intégrale, longueur, aire, to the Sorbonne under the supervision of Émile Borel, earning his doctorate that year.^[10]^[11]^[12] Lebesgue's subsequent career included positions at the University of Rennes (1902–1906), the University of Poitiers (1906–1910), the Sorbonne (1910–1921), and the Collège de France (1921–1941), where he focused on analysis and geometry until his death on July 26, 1941.^[10]^[13] Lebesgue's integral was profoundly influenced by earlier mathematicians addressing limitations in integration and set theory. Bernhard Riemann's 1854 definition of the integral, while elegant for continuous functions, struggled with discontinuities and lacked tools for handling limits inside integrals. In the 1890s, Camille Jordan extended Riemann's ideas with the concept of "content" for sets, providing a measure-like notion for Jordan-measurable sets, which allowed integration of bounded functions over finite intervals but still faltered on highly discontinuous cases.^[14] Similarly, Giuseppe Peano's work on the content of point sets around 1880–1890 laid groundwork for measuring irregular sets, influencing Lebesgue's approach to non-Riemannian integrability. Building directly on these, Émile Borel's 1898 Leçons sur la théorie des fonctions introduced a rigorous theory of measure for sets of real numbers, emphasizing countable unions and geometric intuition, which Lebesgue adapted to define integration over measurable sets.^[15] The primary motivation for Lebesgue's integral stemmed from the Riemann integral's inadequacies in addressing real analysis problems of the era, particularly in handling discontinuous functions, integration over infinite domains, and interchanging limits with integrals. Riemann integration required functions to be bounded and continuous almost everywhere on finite intervals, excluding examples like the Dirichlet function (1 on rationals, 0 on irrationals), which is discontinuous everywhere yet intuitively integrable to 0.^[4] Over unbounded domains, such as the real line, Riemann sums become unwieldy without proper convergence guarantees. Most critically, Riemann failed to justify term-by-term integration of series or limits of integrals, a persistent issue in Fourier analysis where partial sums of coefficients needed validation.^[14] Lebesgue sought a general theory preserving geometric notions of length, area, and volume while resolving these via measure, enabling integration of broader function classes including those unbounded or defined on infinite sets.^[12] Early applications of the Lebesgue integral demonstrated its power in resolving longstanding problems. In Fourier series, Lebesgue's framework facilitated the Riesz–Fischer theorem (1907), proving that every square-integrable function on [0, 2π] has a Fourier series converging in L² norm, overcoming Riemann's limitations on pointwise convergence. Similarly, in potential theory, Lebesgue applied his integral to study harmonic functions and potentials over irregular domains, providing tools to handle discontinuities in electrostatics and gravitation models that Riemann methods could not adequately address.^[14] These applications quickly established the integral's superiority, influencing subsequent developments in functional analysis and probability.

Intuitive comparison to Riemann integral

The Riemann integral defines the integral of a function over an interval by partitioning the domain into subintervals, selecting tags (points) within each subinterval, forming Riemann sums as the product of subinterval lengths and function values at the tags, and taking the limit as the maximum subinterval length approaches zero.^[16] This approach relies on the continuity or bounded variation of the function to ensure the upper and lower sums converge to the same value, limiting its applicability to functions with too many discontinuities.^[17] In contrast, the Lebesgue integral shifts the partitioning to the range of the function, approximating the integral by slicing the graph horizontally: for each value in the range, it identifies the set of points in the domain where the function attains values near that level, multiplies the "height" by the measure (a generalization of length) of that set, and sums these contributions.^[18] This method, which approximates functions via simple step functions aligned with level sets, enables integration over arbitrary measurable sets rather than just finite unions of intervals, providing a more flexible framework for handling complex domains.^[19] A classic example illustrating this difference is the Dirichlet function on the interval [0,1], defined as f(x) = 1 if x is rational and f(x) = 0 if x is irrational. This function is nowhere continuous and not Riemann integrable, as every partition yields upper sums of 1 and lower sums of 0, preventing convergence.^[20] However, it is Lebesgue integrable: the set where f(x) = 1 (the rationals) has Lebesgue measure zero, while the set where f(x) = 0 (the irrationals) has measure 1, so the integral equals $0 \cdot 1 + 1 \cdot 0 = 0.^[18] The Lebesgue approach offers key advantages, including the ability to interchange limits and integrals for pointwise convergent sequences under milder conditions, as seen in theorems like monotone convergence, which fail for Riemann integrals.^[21] Additionally, it naturally supports Fubini's theorem for computing multiple integrals over product measures, avoiding issues with order of integration that can arise in Riemann settings.^[17]

Foundations

Measure spaces and measures

A measure space is a triple (X, \Sigma, \mu), where X is a nonempty set, \Sigma is a \sigma-algebra of subsets of X, and \mu: \Sigma \to [0, \infty] is a measure on \Sigma.^[22] The \sigma-algebra \Sigma consists of the measurable sets, which are closed under countable unions, countable intersections, and complements relative to X.^[23] The measure \mu assigns a nonnegative extended real number to each measurable set, quantifying its "size" in a way that generalizes notions like length, area, and volume.^[24] A measure \mu satisfies \mu(\emptyset) = 0 and countable additivity: for any countable collection of pairwise disjoint sets \{E_k\}_{k=1}^\infty \subset \Sigma, \mu\left(\bigcup_{k=1}^\infty E_k\right) = \sum_{k=1}^\infty \mu(E_k).^[22] This \sigma-additivity implies finite additivity for finite disjoint unions.^[24] From countable additivity, measures inherit monotonicity: if A \subset B with A, B \in \Sigma, then \mu(A) \leq \mu(B).^[24] Measures also exhibit countable subadditivity: for any countable collection \{E_k\}_{k=1}^\infty \subset \Sigma, \mu\left(\bigcup_{k=1}^\infty E_k\right) \leq \sum_{k=1}^\infty \mu(E_k). The Lebesgue measure on \mathbb{R}^n is constructed using an outer measure \mu^* defined for any set E \subset \mathbb{R}^n as the infimum of sums of volumes of countable coverings of E by open rectangles (or cubes).^[25] Specifically, \mu^*(E) = \inf \left\{ \sum_{k=1}^\infty |I_k| : E \subset \bigcup_{k=1}^\infty I_k, \, I_k \text{ open rectangles} \right\}, where |I_k| denotes the volume of I_k.^[26] A set E \subset \mathbb{R}^n is Lebesgue measurable if it satisfies Carathéodory's criterion: for every A \subset \mathbb{R}^n, \mu^*(A) = \mu^*(A \cap E) + \mu^*(A \setminus E).^[25] The collection of Lebesgue measurable sets forms a \sigma-algebra, and the restriction of \mu^* to this \sigma-algebra yields the Lebesgue measure \lambda, which is countably additive on measurable sets.^[26] Examples of measures include the Lebesgue measure \lambda on \mathbb{R}, which assigns to each interval (a,b) the length b-a and extends to all Borel sets via the above construction.^[25] The counting measure on a set X is defined by \mu(E) = |E| if E is finite and \infty otherwise, for E \subset X.^[22] The Dirac measure \delta_x at a point x \in X is given by \delta_x(E) = 1 if x \in E and $0 otherwise, serving as a point mass.^[22] A null set in a measure space (X, \Sigma, \mu) is a measurable set N \in \Sigma with \mu(N) = 0.^[27] Subsets of null sets may not be measurable unless the measure is complete, but in complete measure spaces like the Lebesgue space, all subsets of null sets are null and measurable.^[22] Two functions f, g: X \to \mathbb{R} (or sets) are equal almost everywhere if \{x \in X : f(x) \neq g(x)\} is contained in a null set, meaning they agree except on a set of measure zero.^[27] This equivalence relation is fundamental, as measures disregard differences on null sets.^[24]

Measurable functions and simple functions

In a measure space (X, \Sigma, \mu), a function f: X \to \overline{\mathbb{R}} (where \overline{\mathbb{R}} denotes the extended real numbers) is called measurable if the preimage f^{-1}(B) \in \Sigma for every Borel set B \subseteq \mathbb{R}.^[28] Equivalently, f is measurable if \{x \in X : f(x) > a\} \in \Sigma for all a \in \mathbb{R}, or if the sets where f takes values in intervals like (-\infty, a], (a, b], or (a, \infty) belong to \Sigma.^[29] This definition ensures that measurable functions respect the structure of the \sigma-algebra \Sigma, making them suitable for integration with respect to the measure \mu.^[30] Measurable functions exhibit closure under various operations. The pointwise sum, product, and absolute value of measurable functions are measurable, as are limits of sequences of measurable functions.^[28] Compositions with continuous functions preserve measurability: if f is measurable and g: \overline{\mathbb{R}} \to \overline{\mathbb{R}} is continuous, then g \circ f is measurable.^[29] Additionally, if two functions f and g agree almost everywhere (i.e., \mu(\{x : f(x) \neq g(x)\}) = 0), then f is measurable if and only if g is measurable.^[30] These properties facilitate the analysis of limits and modifications in integration contexts. Simple functions serve as fundamental building blocks for measurable functions in Lebesgue integration. A simple function \phi: X \to \mathbb{R} is a finite linear combination \phi = \sum_{i=1}^n c_i \mathbf{1}_{E_i}, where each c_i \in \mathbb{R}, the E_i \subseteq X are measurable sets (typically taken disjoint for a canonical representation), and \mathbf{1}_{E_i} denotes the indicator function of E_i.^[28] The indicator function \mathbf{1}_E(x) equals 1 if x \in E and 0 otherwise, for a measurable set E \in \Sigma; its integral over X is defined as \mu(E), providing the measure of the set directly.^[29] Simple functions are themselves measurable, as they are finite combinations of measurable indicator functions.^[30] A key approximation result states that every non-negative measurable function f: X \to [0, \infty] can be expressed as the pointwise limit of an increasing sequence of simple functions \{\phi_n\}_{n=1}^\infty such that \phi_n \uparrow f as n \to \infty.^[28] To construct such a sequence, partition the range of f into intervals and define \phi_n(x) = \sum_{k=0}^{n 2^n - 1} \frac{k}{2^n} \mathbf{1}_{E_{k,n}}(x) + n \mathbf{1}_{F_n}(x), where the E_{k,n} capture levels up to \frac{k}{2^n} and F_n handles values exceeding n; this ensures \phi_n \leq f and \phi_n(x) \to f(x) for all x. This theorem underpins the extension of integration from simple functions to general non-negative measurable functions.^[30]

Definition

Integration of non-negative measurable functions

The Lebesgue integral of a non-negative simple function is defined as follows. A simple function \phi on a measure space (X, \mathcal{A}, \mu) takes the form \phi = \sum_{i=1}^n c_i \chi_{E_i}, where each c_i \geq 0 is a constant, the E_i are disjoint measurable sets whose union is X, and \chi_{E_i} is the characteristic function of E_i. The integral is then \int_X \phi \, d\mu = \sum_{i=1}^n c_i \mu(E_i), with the understanding that the sum may be infinite if some \mu(E_i) = \infty. This definition extends the classical notion of integration for step functions to general measure spaces.^[12] For a general non-negative measurable function f: X \to [0, \infty], the Lebesgue integral is defined as the supremum of the integrals of simple functions approximating f from below:

\int_X f \, d\mu = \sup \left\{ \int_X \phi \, d\mu : \phi \text{ simple}, \, 0 \leq \phi \leq f \right\}.

This construction ensures that every non-negative measurable function has a well-defined integral, which may take values in [0, \infty]. The definition relies on the fact that non-negative measurable functions can be approximated by increasing sequences of simple functions, though the supremum captures the essential limit without requiring convergence theorems at this stage.^[31] The integral possesses linearity for non-negative simple functions: if \phi and \psi are simple with \phi, \psi \geq 0 and a, b \geq 0, then \int_X (a\phi + b\psi) \, d\mu = a \int_X \phi \, d\mu + b \int_X \psi \, d\mu. This follows directly from the finite sum representation, as linear combinations of simple functions remain simple. Monotonicity holds for non-negative measurable functions: if $0 \leq f \leq g almost everywhere, then \int_X f \, d\mu \leq \int_X g \, d\mu, since any simple approximant to f is also an approximant to g, yielding a smaller or equal supremum. Additionally, the integral of the zero function satisfies \int_X 0 \, d\mu = 0, providing a baseline for the theory.^[32] To integrate over a measurable subset E \in \mathcal{A}, the restricted integral is defined as \int_E f \, d\mu = \int_X f \chi_E \, d\mu, where \chi_E is the characteristic function of E. Since f \chi_E is non-negative and measurable whenever f is, this inherits the previous definition and properties, allowing computation over subspaces while preserving additivity for disjoint sets. Infinite values are handled naturally: if f > 0 on a set of positive measure and \mu(X) = \infty, the integral may diverge to \infty.^[31]

Extension to signed and complex functions

To extend the Lebesgue integral from non-negative measurable functions to signed real-valued measurable functions on a measure space (X, \mathcal{A}, \mu), a measurable function f: X \to \mathbb{R} is decomposed into its positive and negative parts: f^+(x) = \max(f(x), 0) and f^-(x) = -\min(f(x), 0). This Jordan decomposition yields f = f^+ - f^- and |f| = f^+ + f^-, where both f^+ and f^- are non-negative measurable functions. The integral is defined as \int_X f \, d\mu = \int_X f^+ \, d\mu - \int_X f^- \, d\mu, provided that both \int_X f^+ \, d\mu < \infty and \int_X f^- \, d\mu < \infty; if either is infinite, the integral of f is undefined. A signed measurable function f is said to be (Lebesgue) integrable if \int_X |f| \, d\mu < \infty, in which case both positive and negative parts have finite integrals, ensuring the integral of f exists and is finite. The collection of all such integrable functions, denoted L^1(X, \mu) or simply L^1(\mu), forms a vector space over \mathbb{R} under pointwise addition and scalar multiplication, with the integral serving as a linear functional on this space. This absolute integrability condition guarantees that the integral behaves well under limits and operations, distinguishing it from mere differences of potentially infinite integrals.^[33] For complex-valued measurable functions f: X \to \mathbb{C}, write f = u + iv where u = \operatorname{Re} f and v = \operatorname{Im} f are the real and imaginary parts, both real-valued measurable functions. The integral is then defined as \int_X f \, d\mu = \int_X u \, d\mu + i \int_X v \, d\mu, provided that f is integrable, i.e., \int_X |f| \, d\mu < \infty where |f| = \sqrt{u^2 + v^2}. This ensures both real and imaginary parts are integrable in the signed sense. The space of complex integrable functions, also denoted L^1(\mu), forms a vector space over \mathbb{C}, and the integral is linear over \mathbb{C}. Additionally, for any integrable complex f, \int_X \overline{f} \, d\mu = \overline{\int_X f \, d\mu}, reflecting the conjugate symmetry of the integral.^[34]

Properties

Monotone convergence and dominated convergence theorems

The monotone convergence theorem is a fundamental result in Lebesgue integration that allows the interchange of limits and integrals for increasing sequences of non-negative functions. Specifically, let (X, \mathcal{A}, \mu) be a measure space, and let \{f_n\}_{n=1}^\infty be a sequence of non-negative measurable functions such that f_n \uparrow f pointwise, where f is also measurable and non-negative. Then, \int_X f_n \, d\mu \uparrow \int_X f \, d\mu.^[35] A proof sketch proceeds by first approximating f from below by an increasing sequence of simple functions \{\phi_k\}, so that \int_X \phi_k \, d\mu \uparrow \int_X f \, d\mu. For each fixed n, since f_n \leq f, there exists k_n such that \phi_{k_n} \geq f_n on a set of finite measure, but the standard approach uses the definition of the integral as a supremum over simple functions and monotonicity to show the integrals converge.^[36] Fatou's lemma provides a weaker form of convergence for non-negative sequences, stating that if \{f_n\}_{n=1}^\infty is a sequence of non-negative measurable functions, then \int_X \liminf_{n \to \infty} f_n \, d\mu \leq \liminf_{n \to \infty} \int_X f_n \, d\mu. This inequality follows from applying the monotone convergence theorem to the sequence g_n = \inf_{k \geq n} f_k, which is increasing to \liminf f_n.^[37] The dominated convergence theorem extends this to signed functions under a uniform integrability condition. Let \{f_n\}_{n=1}^\infty be a sequence of measurable functions converging pointwise to f, and suppose there exists an integrable function g \geq 0 such that |f_n| \leq g \mu-almost everywhere for all n. Then f is integrable, and \int_X f_n \, d\mu \to \int_X f \, d\mu. A proof sketch applies the monotone convergence theorem to the non-negative functions |f_n - f| and |f_n|, using the domination to bound the differences and pass limits inside the integral via Fatou's lemma on $2g - |f_n - f|.^[35] Beppo Levi's theorem is a variant of the monotone convergence theorem specialized to series of non-negative functions. If \{f_n\}_{n=1}^\infty is a sequence of non-negative measurable functions, then \int_X \sum_{n=1}^\infty f_n \, d\mu = \sum_{n=1}^\infty \int_X f_n \, d\mu, where the sum on the left is the integral of the pointwise sum, provided it is measurable. This follows directly from applying the monotone convergence theorem to the partial sums s_m = \sum_{n=1}^m f_n \uparrow \sum_{n=1}^\infty f_n.^[38]

Linearity, inequalities, and differentiation theorems

The Lebesgue integral possesses the linearity property, which states that if f and g are integrable functions on a measure space (X, \mathcal{A}, \mu) and a, b are scalars, then af + bg is integrable and

\int_X (af + bg) \, d\mu = a \int_X f \, d\mu + b \int_X g \, d\mu.

This holds by construction, as the integral is first defined for non-negative simple functions via finite sums, where linearity is immediate, and then extended by limits to general non-negative measurable functions and subsequently to signed functions via decomposition into positive and negative parts.^[1] A fundamental inequality for the Lebesgue integral is the triangle inequality, which asserts that if f is integrable, then

\left| \int_X f \, d\mu \right| \leq \int_X |f| \, d\mu.

This follows directly from the linearity applied to the positive and negative parts of f, since \int_X f \, d\mu = \int_X f^+ \, d\mu - \int_X f^- \, d\mu and \int_X |f| \, d\mu = \int_X f^+ \, d\mu + \int_X f^- \, d\mu, yielding the absolute value bound.^[1] Hölder's inequality provides a more refined bound on products of functions: if $1 < p < \infty, q is the conjugate exponent satisfying \frac{1}{p} + \frac{1}{q} = 1, and f \in L^p(\mu), g \in L^q(\mu), then fg \in L^1(\mu) and

\left| \int_X fg \, d\mu \right| \leq \left( \int_X |f|^p \, d\mu \right)^{1/p} \left( \int_X |g|^q \, d\mu \right)^{1/q}.

This inequality, originally established for integrals in the context of mean values, underpins the duality of L^p spaces and extends the Cauchy-Schwarz case (p = q = 2).^[39] For convex functions, Jensen's inequality adapts to the Lebesgue integral on probability spaces: if \mu(X) = 1, f is integrable, and \phi: \mathbb{R} \to \mathbb{R} is convex, then

\phi\left( \int_X f \, d\mu \right) \leq \int_X \phi(f) \, d\mu.

This reflects the preservation of convexity under integration, with equality holding if \phi is linear on the range of f, and it generalizes to finite measures by normalization.^[40] Fubini's theorem justifies iterated integration over product measures: if f is integrable on the product space (X \times Y, \mathcal{A} \times \mathcal{B}, \mu \times \nu), then for almost every y \in Y, the section x \mapsto f(x,y) is integrable on X, the function y \mapsto \int_X f(x,y) \, d\mu(x) is integrable on Y, and

\int_Y \left( \int_X f(x,y) \, d\mu(x) \right) d\nu(y) = \int_{X \times Y} f \, d(\mu \times \nu) = \int_X \left( \int_Y f(x,y) \, d\nu(y) \right) d\mu(x).

This allows computation of multiple integrals by successive single integrals under the integrability condition. The Leibniz rule for differentiation under the Lebesgue integral applies to parameter-dependent integrands: suppose f: I \times X \to \mathbb{R} (with I an interval containing t_0) is such that for t near t_0, the partial derivative \frac{\partial}{\partial t} f(t,x) exists for almost every x \in X, t \mapsto f(t,x) is continuous at t_0, and there exists an integrable g: X \to [0,\infty) dominating |\frac{\partial}{\partial t} f(t,x)| for t near t_0. Then t \mapsto \int_X f(t,x) \, d\mu(x) is differentiable at t_0 with

\frac{d}{dt} \int_X f(t,x) \, d\mu(x) \bigg|_{t=t_0} = \int_X \frac{\partial}{\partial t} f(t,x) \, d\mu(x) \bigg|_{t=t_0}.

The domination ensures the interchange of limit and integral via the dominated convergence theorem.^[41]

Examples and Applications

Elementary examples in one dimension

One of the simplest examples of a Lebesgue integral arises from step functions, which are finite linear combinations of indicator functions of intervals. Consider the Lebesgue measure \lambda on \mathbb{R}, which assigns to each interval its length. For the step function \phi(x) = 1 if $0 \leq x < 1/2 and \phi(x) = 0 otherwise on [0,1], the Lebesgue integral is computed as \int_{[0,1]} \phi \, d\lambda = 1 \cdot \lambda([0,1/2)) + 0 \cdot \lambda([1/2,1]) = 1/2.^[42] This matches the Riemann integral, as \phi is piecewise continuous. To illustrate the definition more generally, consider approximating the indicator function of an interval using simple functions. The function f(x) = \mathbf{1}_{[0,1]}(x) is itself simple, with \int_{\mathbb{R}} f \, d\lambda = \lambda([0,1]) = 1. For approximation, one can construct a sequence of simple functions \phi_n as unions of dyadic intervals, such as \phi_n(x) = \sum_{k=0}^{2^n - 1} \mathbf{1}_{[k/2^n, (k+1)/2^n]}(x) for x \in [0,1), where each \phi_n integrates to 1 and converges pointwise to f.^[43] This process highlights how the Lebesgue integral extends to more complex functions via limits of simple approximations. A key advantage of the Lebesgue integral over the Riemann integral appears with discontinuous functions like the Dirichlet function on [0,1], defined as d(x) = 1 if x is rational and d(x) = 0 if x is irrational. The set of rationals in [0,1] has Lebesgue measure zero, so d(x) = 0 almost everywhere, and \int_{[0,1]} d \, d\lambda = 0.^[44] In contrast, the Dirichlet function is nowhere continuous and thus not Riemann integrable on [0,1]. Thomae's function, also known as the popcorn function, on [0,1] is defined as t(x) = 1/q if x = p/q in lowest terms with q > 0, and t(x) = 0 if x is irrational. This function is discontinuous precisely at the rationals, a set of measure zero, making it bounded with discontinuities on a null set. Consequently, it is Riemann integrable with \int_0^1 t(x) \, dx = 0, and thus also Lebesgue integrable with the same value, despite the discontinuities at every rational point.^[16]^[44] For improper integrals over unbounded domains, the monotone convergence theorem facilitates computation. Consider f(x) = e^{-x} for x \geq 0 and $0 otherwise. Define partial functions f_n(x) = e^{-x} \mathbf{1}_{[0,n]}(x), which increase pointwise to f(x) and are integrable. Each \int_{\mathbb{R}} f_n \, d\lambda = 1 - e^{-n}, so by monotone convergence, \int_0^\infty e^{-x} \, d\lambda = \lim_{n \to \infty} (1 - e^{-n}) = 1.^[45] This aligns with the improper Riemann integral but leverages the Lebesgue framework for monotone sequences.

Applications in probability and analysis

In probability theory, the Lebesgue integral serves as the rigorous foundation for defining the expectation of a random variable. For a random variable X defined on a probability space (\Omega, \mathcal{F}, P) with P(\Omega) = 1, the expectation is expressed as E[X] = \int_{\Omega} X \, dP, which aligns with the Lebesgue integral with respect to the probability measure P. This formulation ensures that expectations can be computed for a broad class of random variables, including those that are not continuous or bounded, by approximating via simple functions and leveraging measure-theoretic properties.^[46] The approach extends to higher-order moments, such as variance and skewness, through the L^p spaces where E[|X|^p] = \int_{\Omega} |X|^p \, dP < \infty for $1 \leq p < \infty, providing a unified framework for analyzing moment conditions in probabilistic models.^[47] The Fubini-Tonelli theorem further enhances applications in probability by justifying the interchange of integrals for joint distributions. For non-negative measurable functions on product probability spaces, it permits evaluating double integrals as iterated expectations, such as \int_{\Omega_1 \times \Omega_2} f(x,y) \, dP_1(x) dP_2(y) = \int_{\Omega_1} \left( \int_{\Omega_2} f(x,y) \, dP_2(y) \right) dP_1(x), which is crucial for computing covariances or expectations of products of independent random variables. This result underpins key identities like E[XY] = E[X]E[Y] for independent X and Y, facilitating the analysis of multivariate distributions without restrictions to finite support. In functional analysis, the Lebesgue integral defines the L^p(\mu) spaces for $1 \leq p \leq \infty on a measure space (\Omega, \mathcal{F}, \mu), equipped with the norm \|f\|_p = \left( \int_{\Omega} |f|^p \, d\mu \right)^{1/p} (with the essential supremum for p = \infty). These spaces form Banach spaces, meaning they are complete normed vector spaces, a property proven by showing that Cauchy sequences converge in the L^p norm using the dominated convergence theorem to pass limits inside the integral. The completeness ensures that L^p spaces support advanced techniques like fixed-point theorems and operator theory, essential for solving partial differential equations and variational problems.^[48] Fourier analysis relies on the Lebesgue integral to establish Plancherel's theorem, which preserves the inner product structure in L^2(\mathbb{R}). Specifically, for f, g \in L^2(\mathbb{R}), the theorem asserts \int_{\mathbb{R}} f(x) \overline{g(x)} \, dx = \int_{\mathbb{R}} \hat{f}(\xi) \overline{\hat{g}(\xi)} \, d\xi, where \hat{f} denotes the Fourier transform, confirming that the Fourier transform is an isometry on L^2. This equality, derived through density arguments and Lebesgue integration, enables the Parseval identity for energy preservation in signal processing and quantum mechanics, where Riemann integrals fail for non-absolutely integrable functions.^[49] Post-2000 developments have extended Lebesgue integral applications to stochastic processes, particularly for handling high-dimensional measures. In stochastic calculus, the Lebesgue integral underpins the definition of Itô integrals for semimartingales, allowing integration against paths of processes like Brownian motion in financial modeling and diffusion simulations, as detailed in modern treatments of stochastic differential equations.^[50]

Limitations and Extensions

Shortcomings relative to Riemann integral

While every bounded function on a closed interval [a, b] that is Riemann integrable is also Lebesgue integrable, with the two integrals coinciding, the converse does not hold. A bounded function f: [a, b] \to \mathbb{R} is Riemann integrable if and only if it is continuous almost everywhere with respect to Lebesgue measure. This equivalence highlights that the Riemann integral captures a subset of Lebesgue-integrable functions, specifically those without too many discontinuities.^[51]^[52] A classic example of a function that is Lebesgue integrable but not Riemann integrable is the characteristic function \chi_{\mathbb{Q}} of the rational numbers on [0, 1], which equals 1 on rationals and 0 on irrationals. This function is discontinuous everywhere, so its upper and lower Riemann sums never converge to the same value, rendering it non-Riemann integrable. However, since the rationals have Lebesgue measure zero, \chi_{\mathbb{Q}} is equal almost everywhere to the zero function, which is Lebesgue integrable with integral 0.^[18] The Lebesgue integral fails entirely for non-measurable functions, such as the Vitali set V \subset [0, 1], a standard example of a non-Lebesgue measurable set. The characteristic function \chi_V cannot be Lebesgue integrated because V lacks a well-defined Lebesgue measure; assigning it a measure would lead to a contradiction with the additivity of Lebesgue measure over disjoint translates of V. The construction of the Vitali set relies on the axiom of choice to select one representative from each equivalence class of \mathbb{R}/\mathbb{Q}, underscoring that non-measurable sets—and thus certain integration failures—depend on this axiom. From a computational perspective, numerical approximation of the Lebesgue integral can be more involved than for the Riemann integral, particularly for functions requiring level set or occupation time discretizations, which may exhibit different convergence properties compared to standard Riemann sums.^[53] The Riemann integral is inherently restricted to functions defined on bounded intervals, limiting its applicability to more general domains, whereas the Lebesgue integral operates over abstract measure spaces. Nevertheless, for continuous functions on compact intervals, the Riemann integral remains simpler and more direct, as uniform continuity ensures straightforward convergence of Riemann sums without needing measure-theoretic machinery.^[52]

Alternative formulations and generalizations

One alternative formulation of the Lebesgue integral for non-negative measurable functions f on a measure space (X, \mathcal{M}, \mu) defines the upper integral as the supremum over all simple functions \phi \leq f of \int \phi \, d\mu, and the lower integral as the infimum over all simple functions \psi \geq f of \int \psi \, d\mu; the function is integrable if these coincide, yielding the integral value.^[22] This approach parallels the Riemann upper and lower sums but replaces partitions with measurable sets and lengths with measures, ensuring the integral equals the common value for measurable f.^[22] Unlike the Riemann case, where non-equality implies non-integrability, the Lebesgue version guarantees equality for all measurable functions due to the density of simple functions in the measurable ones.^[22] The Daniell integral provides an abstract construction starting from a vector space of functions, such as continuous functions on a topological space, equipped with a positive linear functional I_0 that satisfies monotonicity and normalization properties.^[54] This functional is extended to a larger space of lower semi-continuous functions by defining the upper integral as the infimum of I_0 over majorants, yielding a complete integration theory without explicit measure construction.^[54] On \mathbb{R}^n with the standard topology, the Daniell integral coincides with the Lebesgue integral, as shown by the Daniell-Stone theorem, which links it to regular Borel measures via the Riesz representation theorem.^[55] The Henstock-Kurzweil integral, also known as the gauge integral, generalizes the Perron integral by refining Riemann sums with a gauge function \delta: [a,b] \to (0,\infty), where partitions are admissible if each subinterval length is less than the gauge value at its tag point. A bounded function f on [a,b] is integrable if there exists L \in \mathbb{R} such that for every \varepsilon > 0, there is a gauge \delta where the difference between any admissible Riemann sum and L is at most \varepsilon. On \mathbb{R}^n, it recovers the Lebesgue integral for Lebesgue measurable functions and additionally integrates all continuous functions and all derivatives of continuous functions, which may not be Lebesgue integrable. Generalizations of the Lebesgue integral extend it to broader settings. The Bochner integral applies to functions f: X \to B with values in a Banach space B, defining \int f \, d\mu as the limit of integrals of simple functions \sum \mu(E_i) v_i approximating f, where f is Bochner integrable if \int \|f\| \, d\mu < \infty.^[56] This preserves linearity and convergence theorems like monotone and dominated convergence in the vector-valued case.^[56] For product spaces, Fubini's theorem allows iterated integration: if f is integrable on the product measure space (X \times Y, \mu \times \nu), then \int_X \left( \int_Y |f(x,y)| \, d\nu(y) \right) d\mu(x) = \int_{X \times Y} |f| \, d(\mu \times \nu) < \infty, and the double integral equals the iterated ones in either order.^[57] Abstract integration via Haar measure generalizes Lebesgue integration to locally compact topological groups G, where a left-invariant Borel measure m (Haar measure) exists, unique up to scalar multiple, and is finite on compact sets.^[58] Integration with respect to m defines \int_G f \, dm for integrable f: G \to \mathbb{C}, extending Lebesgue integration on \mathbb{R}^n (where Lebesgue measure is the Haar measure) and applying to non-σ-finite spaces like infinite discrete groups with counting measure.^[58] This framework supports group-invariant analysis, such as Fourier transforms on non-abelian groups.^[58]

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