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Introduction

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The statement of the theorem

The integral of a positive real function f between boundaries a and b can be interpreted as the area under the graph of f, between a and b. This notion of area fits some functions, mainly piecewise continuous functions, including elementary functions, for example polynomials. However, the graphs of other functions, for example the Dirichlet function, do not fit well with the notion of area. Graphs like that of the latter, raise the question: for which class of functions does "area under the curve" make sense? The answer to this question has great theoretical importance. As part of a general movement toward rigor in mathematics in the nineteenth century, mathematicians attempted to put integral calculus on a firm foundation. The Riemann integral—proposed by Bernhard Riemann (1826–1866)—is a broadly successful attempt to provide such a foundation. Riemann's definition starts with the construction of a sequence of easily calculated areas that converge to the integral of a given function. This definition is successful in the sense that it gives the expected answer for many already-solved problems, and gives useful results for many other problems. However, Riemann integration does not interact well with taking limits of sequences of functions, making such limiting processes difficult to analyze. This is important, for instance, in the study of Fourier series, Fourier transforms, and other topics. The Lebesgue integral describes better how and when it is possible to take limits under the integral sign (via the monotone convergence theorem and dominated convergence theorem). While the Riemann integral considers the area under a curve as made out of vertical rectangles, the Lebesgue definition considers horizontal slabs that are not necessarily just rectangles, and so it is more flexible. For this reason, the Lebesgue definition makes it possible to calculate integrals for a broader class of functions. For example, the Dirichlet function, which is 1 where its argument is rational and 0 otherwise, has a Lebesgue integral, but does not have a Riemann integral. Furthermore, the Lebesgue integral of this function is zero, which agrees with the intuition that when picking a real number uniformly at random from the unit interval, the probability of picking a rational number should be zero. Lebesgue summarized his approach to integration in a letter to Paul Montel: I have to pay a certain sum, which I have collected in my pocket. I take the bills and coins out of my pocket and give them to the creditor in the order I find them until I have reached the total sum. This is the Riemann integral. But I can proceed differently. After I have taken all the money out of my pocket I order the bills and coins according to identical values and then I pay the several heaps one after the other to the creditor. This is my integral. — Source: (Siegmund-Schultze 2008) The insight is that one should be able to rearrange the values of a function freely, while preserving the value of the integral. This process of rearrangement can convert a very pathological function into one that is "nice" from the point of view of integration, and thus let such pathological functions be integrated. A measurable function is shown, together with the set {x : f(x) > t} (on the x-axis). The Lebesgue integral is obtained by slicing along the y-axis, using the 1-dimensional Lebesgue measure to measure the "width" of the slices. Folland (1999) summarizes the difference between the Riemann and Lebesgue approaches thus: "to compute the Riemann integral of f, one partitions the domain [a, b] into subintervals", while in the Lebesgue integral, "one is in effect partitioning the range of f". For the Riemann integral, the domain is partitioned into intervals, and bars are constructed to meet the height of the graph. The areas of these bars are added together, and this approximates the integral, in effect by summing areas of the form f(x)dx where f(x) is the height of a rectangle and dx is its width. For the Lebesgue integral, the range is partitioned into intervals, and so the region under the graph is partitioned into horizontal "slabs" (which may not be connected sets). The area of a small horizontal "slab" under the graph of f, of height dy, is equal to the measure of the slab's width times dy: μ({x∣f(x)>y}) dy.\mu \left(\{x\mid f(x)>y\}\right)\,dy. The Lebesgue integral may then be defined by adding up the areas of these horizontal slabs. From this perspective, a key difference with the Riemann integral is that the "slabs" are no longer rectangular (cartesian products of two intervals), but instead are cartesian products of a measurable set with an interval. Riemannian (top) vs. Lebesgue (bottom) integration of smoothed COVID-19 daily case data from Serbia (Summer-Fall 2021) An equivalent way to introduce the Lebesgue integral is to use so-called simple functions, which generalize the step functions of Riemann integration. Consider, for example, determining the cumulative COVID-19 case count from a graph of smoothed cases each day (right). The Riemann–Darboux approach Partition the domain (time period) into intervals (eight, in the example) and construct bars with heights that meet the graph. The cumulative count is found by summing, over all bars, the product of interval width (time in days) and the bar height (cases per day).The Lebesgue approach Choose a finite number of target values (eight, in the example) in the range of the function. By constructing bars with heights equal to these values, but below the function, they imply a partitioning of the domain into the same number of subsets (subsets, indicated by color in the example, need not be connected). This is a "simple function", as described below. The cumulative count is found by summing, over all subsets of the domain, the product of the measure on that subset (total time in days) and the bar height (cases per day). One can think of the Lebesgue integral either in terms of slabs or simple functions. Intuitively, the area under a simple function can be partitioned into slabs based on the (finite) collection of values in the range of a simple function (a real interval). Conversely, the (finite) collection of slabs in the undergraph of the function can be rearranged after a finite repartitioning to be the undergraph of a simple function. The slabs viewpoint makes it easy to define the Lebesgue integral, in terms of basic calculus. Suppose that f is a (Lebesgue measurable) function, taking non-negative values (possibly including +∞). Define the distribution function of f as the "width of a slab", i.e., F(y)=μ{x∣f(x)>y}.F(y)=\mu \{x|f(x)>y\}. Then F(y) is monotone decreasing and non-negative, and therefore has an (improper) Riemann integral over (0, ∞), allowing that the integral can be +∞. The Lebesgue integral can then be defined by ∫f dμ=∫0∞F(y) dy\int f\,d\mu =\int _{0}^{\infty }F(y)\,dy where the integral on the right is an ordinary improper Riemann integral, of a non-negative function (interpreted appropriately as +∞ if F(y) = +∞ on a neighborhood of 0). Most textbooks, however, emphasize the simple functions viewpoint, because it is then more straightforward to prove the basic theorems about the Lebesgue integral. Measure theory was initially created to provide a useful abstraction of the notion of length of subsets of the real line—and, more generally, area and volume of subsets of Euclidean spaces. In particular, it provided a systematic answer to the question of which subsets of R have a length. As later set theory developments showed (see non-measurable set), it is actually impossible to assign a length to all subsets of R in a way that preserves some natural additivity and translation invariance properties. This suggests that picking out a suitable class of measurable subsets is an essential prerequisite. The Riemann integral uses the notion of length explicitly. Indeed, the element of calculation for the Riemann integral is the rectangle [a, b] × [c, d], whose area is calculated to be (b − a)(d − c). The quantity b − a is the length of the base of the rectangle and d − c is the height of the rectangle. Riemann could only use planar rectangles to approximate the area under the curve, because there was no adequate theory for measuring more general sets. In the development of the theory in most modern textbooks (after 1950), the approach to measure and integration is axiomatic. This means that a measure is any function μ defined on a certain class X of subsets of a set E, which satisfies a certain list of properties. These properties can be shown to hold in many different cases. We start with a measure space(E, X, μ) where E is a set, X is a σ-algebra of subsets of E, and μ is a (non-negative) measure on E defined on the sets of X. For example, E can be Euclidean n-spaceR^{n} or some Lebesgue measurable subset of it, X is the σ-algebra of all Lebesgue measurable subsets of E, and μ is the Lebesgue measure. In the mathematical theory of probability, we confine our study to a probability measure μ, which satisfies μ(E) = 1. Lebesgue's theory defines integrals for a class of functions called measurable functions. A real-valued function f on E is measurable if the pre-image of every interval of the form (t, ∞) is in X: {x ∣ f(x)>t}∈X∀t∈R.\{x\,\mid \,f(x)>t\}\in X\quad \forall t\in \mathbb {R} . We can show that this is equivalent to requiring that the pre-image of any Borel subset of R be in X. The set of measurable functions is closed under algebraic operations, but more importantly it is closed under various kinds of point-wise sequential limits: sup⁡k∈Nfk,lim inf⁡k∈Nfk,lim sup⁡k∈Nfk\sup _{k\in \mathbb {N} }f_{k},\quad \liminf _{k\in \mathbb {N} }f_{k},\quad \limsup _{k\in \mathbb {N} }f_{k} are measurable if the original sequence (f_{k}), where k ∈ N, consists of measurable functions. There are several approaches for defining an integral for measurable real-valued functions f defined on E, and several notations are used to denote such an integral. ∫Ef dμ=∫Ef(x) dμ(x)=∫Ef(x) μ(dx).\int _{E}f\,d\mu =\int _{E}f(x)\,d\mu (x)=\int _{E}f(x)\,\mu (dx). Following the identification in Distribution theory of measures with distributions of order 0, or with Radon measures, one can also use a dual pair notation and write the integral with respect to μ in the form ⟨μ,f⟩.\langle \mu ,f\rangle .